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Does “Second X is NPcomplete” imply “X is NPcomplete”?
"Second $X$" problem is the problem of deciding the existence of another solution different from some given solution for problem instance.
For some $NP$complete problems, the second solution version is $NP$complete (deciding the existence of another solution for the partial Latin square completion problem) while for others it is either trivial (Second NAE SAT) or it can not be $NP$complete (Second Hamiltonian cycle in cubic graphs) under widely believed complexity conjecture. I am interested in the opposite direction.
We assume a natural $NP$ problem $X$ where there is natural efficient verifier that verifies a natural interesting relation $(x, c)$ where $x$ is an input instance and $c$ is a short witness of membership of $x$ in $X$. All witnesses are indistinguishable to the verifier. The validity of witnesses must be decided by running the natural verifier and it does not have any knowledge of any correct witness ( both examples in the comments are solutions by d

The answer is yes (if ASP reduction is used instead of Karp reduction). ASP reduction requires a polynomial time computable bijection between the solution sets of the two problems. This provides a parsimonious reduction between ASPcomplete problems. Yato and Seta state that $ASP$completeness imply $NP$completeness (Page 2, second paragraph). Another solution problem (ASP) is exactly what I call Second X problem.
Oded Goldreich states the fact that "all known reductions among natural $NP$complete problems are either parsimonious or can be easily modified to be so". ( Computational Complexity: A Conceptual Perspective By Oded Goldreich). Therefore, it is plausible that Karp reductions between natural NPcomplete problems can be modified to be ASP reductions.
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