Probably not.
Parents are generally always named Joint Managing Conservators (JMC) unless one parent can provide evidence that proves that a parent's immigration status has a material, adverse effect on their ability to parent.
In the recent case of Turrubiartes v. Olvera, 2018 Tex. App. LEXIS 1017 (Tex. App.--Houston [1st Dist.] Februrary 6, 2018)(opinion on rhr'g)(Cause No. 01-16-00322-CV). A father argued that the main reason he should be name sole managing conservator of his three children was that he was a U.S. Citizen and the mother was not. Father said he feared that the mother would be stopped while driving the kids in Texas and be deported.
The lower court granted Sole Managing Conservatorship to the father. In it's findings of fact on the ruling, nine out of the thirteen factors the court said it considered had to with the mother's immigration status.
The Court of Appeal for the 1st District overturned the ruling. They pointed out that "immigration status" is not expressly listed in TFC 153.134(a) as one of the factors that a court may use in overcoming the presumption that parents should be named Joint Managing Conservators. The COA said the father's fear of the mother being deported while driving the children was resolve by her being ordered by the lower court to find a licensed driver for the children. The Court of Appeals concluded that father's other points for him being named sole managing conservator were not sufficient to overcome the joint managing presumption.
Comment:
This is not the last we will hear of immigration status being used as a weapon in a custody case. With the recent Federal Court ruling upholding punitive measures used against so called "sanctuary cities", the political winds still seem to be blowing hard against illegal immigrants in Texas. It should be noted that in this case the COA seemed to ignore the catch all statutory factor under 153.134(a) of "any relevant factor"- which may be used in the future. Future cases, with the right fact pattern in may find success in using immigration status as a reason factor in custody cases. I would not be surprised if a conservative legislator will jump on this and seek to add immigration status to 153.134(a).