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First, given your ex's track record for visitation and custody then you should not be concerned over a Court deciding to give him joint custody. Courts do generally favor as must visitation or shared custody as possible but his lack of really giving an effort through the years does not lend itself to convincing a court to give him more custody.
Second, actually getting a court to take away parental rights and give them to your current husband is very difficult. Generally a court will have to find no communication or visitation for years as well as failure to pay any child support. Even the minimal visitation that your ex has had over the years is likely enough to prevent him from having his parental rights taken away.
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