Let $M$ be a NDTM (non deterministic Turing machine) which decides a certain NP-complete language, say SAT.

 $M$ computes any instance $I$ of the NP-complete problem in at most $p(n)$ non deterministic steps ($p$ in a polynomial function, $n$ is the input size). The length of a computation path can be measured as the "rank" of the non deterministic step where $M$ halts. Let call $rank_M(I)$  this rank ($rank_M(I)$ is $p(n)+1$ if $M$ rejects $I$).

Is this problem NP-complete : Given a boolean expression $I$, is it true that $rank_M(I)=k$ ? ($k$ is polynomially bounded in $n$)


Thank you.