Not very high... joint managing conservatorship is presumed to be in the best interest of the child. Without proof of abandonment (extended periods like 6 months of no support or visitation) or domestic violence, you will likely end up with joint managing conservatorship...
Tx doesn't really use the term 'sole custody' so I assume you mean sole managing conservator.
If he takes the stand and confirms everything above, you might very well get most of the rights outlined in the Texas Family Code, and essentially get what you want. You may want to continue to push that the Wife not be allowed to be part of the visitation...
If you sit down with a family lawyer in Houston, I'm sure they can explain this in more detail and you can make sure to be in the best position by being represented.
Disclaimer: This answer is intended for informational and educational purposes only, and should not be considered legal advice nor forming the attorney-client relationship. This attorney is licensed to practice in Texas and you should always consult an attorney in your jurisdiction before moving forward.
"Sole Custody" is not a term defined by the law in Texas and therefore if means different things to different people.
If you are asking whether you have a chance to terminate the father's rights, the answer is NO.
If you are asking whether you have a chance of obtaining sole managing conservatorship (SMC), the answer is PROBABLY NOT. It is very hard to get SMC from the court due to the strong statutory presumption favoring JMC (joint managing conservatorship).
If you are asking whether you have a chance of being the joint managing conservator with the exclusive right to designate the primary residence of the child, the answer is a strong YES. Please note that even if you are designated as the primary managing conservator, based on the facts you've given, your husband will be able to get an extended standard possession order, if he knows how to ask for it.
If you are asking that in addition to being the primary joint managing conservator you have a chance of holding the lion's share of the rights and duties exclusively, the answer is YES, if you use an attorney. Otherwise, you stand a strong chance that you will not request such relief in a fashion that the court can act upon.
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