Pfaffian multiplicativity (MP) provable in VNC^2

Prove within the bounded arithmetic theory VNC^2 that the Pfaffian multiplicativity identity pf(BAB^t) = pf(A) det(B) holds for all even-dimensional skew-symmetric matrices A and all matrices B, thereby establishing (MP) in VNC^2.

Background

The paper shows that many Pfaffian properties (including alternation and cofactor expansion) follow from the multiplicativity identity (MP): pf(BAB^t) = pf(A) det(B). This parallels the determinant setting where multiplicativity is provable in VNC^2 [9].

Motivated by these parallels, the authors explicitly conjecture that (MP) itself is provable in VNC^2, which would further align the provability of linear-algebraic identities across determinant and Pfaffian frameworks in bounded arithmetic.