Unimodality conjecture for Poincaré polynomials of degenerate flag varieties

For each of the five types considered—type $A$, odd symplectic, even symplectic, odd orthogonal, and even orthogonal—let $P(q)$ denote the corresponding Poincaré polynomial. Unimodality conjecture. The Poincaré polynomials in all five types are unimodal. The conjecture is based on the computed examples and is not proved even in type $A$.