(CNN) — A few dozen immigrant families are expected to be reunited by a court deadline Tuesday, as the fates of thousands more remain in limbo.

Four children of 102 identified for the deadline have already been reunified with their parents, and another 34 were expected to be reunited on Tuesday, according to a court filing Tuesday.

The deadline comes amid heightened stakes. A separate federal judge late Monday flatly rejected the administration’s attempt to rewrite the rules for detaining families in order to hold them indefinitely, calling the request “cynical” and “wholly without merit.”

The children who are addressed under the Tuesday deadline are all under age 5 and have been held by the government for weeks or months after being separated from their parents at the border.

However, the government admits it will be able to meet the deadline only for just over half of them in this group, and they’re just a fraction of the thousands of young immigrants still in custody apart from their parents, many of whom are detained and separated from family as a byproduct of the Trump administration’s “zero-tolerance” policy.

Beyond the 38 who were to be reunited by Tuesday, another 16 children were to be reunited “soon thereafter,” the filing said, as soon as they are confirmed as eligible, and another eight were flagged for follow-up regarding the safety and suitability of the parent.

Twelve more kids have parents who cannot be located yet.

There are 27 children that the government has determined are not eligible for reunification, either because the adult is ineligible due to criminal history or not actually being the parent, or because the parents present a danger to the child or have a contagious disease.

In the filing, the government argued, though, that the numbers still show the government is “in compliance with the court’s order” to meet the Tuesday deadline.

The lawyers and immigrants who have brought this lawsuit notably do not agree that the Trump administration has fully complied with the original deadlines in the judge’s order.

They state they only received the updated numbers “within the past hour” and “have no independent verification that these numbers are accurate, or that there are not additional children under 5 who should be on the government’s list.” The plaintiffs say they “recognize that Defendants cannot yet reunify the parents who are currently being held in criminal custody,” but as for the other parents “with children under five, the government is not in compliance with the clear deadline ordered by the Court.”

In a court hearing Monday, District Judge Dana Sabraw said he was “encouraged” by the “progress,” saying he was confident that “many of these families” will meet the deadline. “Then we’ll have a very clear understanding as to who has not been reunited, why not and what time frame will be in place,” Sabraw added.

New deadlines likely to be set

The attorneys are due back in court Tuesday for an update — and likely to set new deadlines for the roughly half of the children who the government does not expect to reunite on time.

In filings Monday night, the attorneys for the government and for the immigrants represented by the American Civil Liberties Union filed a report for the judge on where they agree and where they don’t. The filing offers some agreement on matters like DNA testing, background checks for parents and limiting home studies, but the judge will have to rule on issues including the extent to which the government investigates parents before their children are released to them.

The main issue is what procedures the government will follow before it reunites families. The government argues it must follow all the steps it would if the child were unaccompanied to ensure the safety of the child.

But the ACLU argues that anything but a streamlined process will result in needless delays, adding that because the government separated the parents and children in the first place, it should expedite their reunion.

“The government should not be allowed to delay reunification to conduct procedures that would not have been used if the child had not been forcibly taken from the parent,” the ACLU wrote in the filing. “If a class member parent and child had showed up at the border together, and had not been separated, then the parent would not be required to undergo the extensive procedures proposed by the Government to maintain custody of the child.”

What about the other immigrant children?

Another key deadline comes at the end of the month. The government has thousands more children 5 and older in its custody who it will have to reunite with parents by July 26, but the Monday hearing did not cover that group. The judge and attorneys did express hope that the efforts to reunify the younger families will help speed up the process for the older group.

The plan for Tuesday’s reunions is for the children to be released to Immigration and Customs Enforcement, which runs the adult detention centers where their parents are being held. Once the children are handed over to their custody, ICE will release the families together. This plan bypasses the lengthy “sponsor” process that is required to release an unaccompanied immigrant child from Health and Human Services custody.

Only about 54 children who have parents currently in government custody are expected to be reunited on Tuesday, though another five who are waiting on unresolved background check issues could come together in time.

The rest largely either have parents already deported, already released somewhere in the US, or in federal or state criminal custody.

The-CNN-Wire™ & © 2018 Cable News Network, Inc., a Time Warner Company. All rights reserved.