Proof of Stasinski and Voll's Hyperoctahedral Group Conjecture

In a recent paper, Stasinski and Voll introduced a length-like statistic on hyperoctahedral groups and conjectured a product formula for this statistic's signed distribution over arbitrary quotients. Stasinski and Voll proved this conjecture for a few special types of quotients. We prove this conjecture in full, showing it holds for all quotients. In the case of signed permutations with at most one descent, this formula gives the Poincare polynomials for the varieties of symmetric matrices of a fixed rank.