Given a $3SAT$ problem the decision version of the problem (if it has a solution or not) is $NPComplete$. Now, given that we consider the promise version of the $3SAT$ problem, the promise is every $3SAT$ problem provided always has at least 1 solution.
Query: Does every solution of the above $3SAT$ problem instance have a particular property (one that can be tested in polynomial time. For eg: 'for any solution $A_i$, $N$ numbers immediately smaller than A are co-prime' or something similar; where $N$ is the input length)?
The problem and property can be encoded as a pair consisting of $3SAT$, and another $N$ input circuit of polynomial size ($N^c$ for some constant $c$)
This promise version is definitely in $co-NP$ (as if there is a solution that does not satisfy the given property, that solution is the $co-NP$ certifcate of polynomial length).
Is this problem also $co-NPComplete$ (as it still involves getting the failing solution first)?