Positivity conjecture for characteristic classes of determinantal varieties

Let $\tau_{n,k}^{\circ}$, $\tau_{n,k}^{\wedge \circ}$, and $\tau_{n,k}^{S \circ}$ denote the corresponding generic determinantal, skew-symmetric determinantal, and symmetric determinantal varieties, and let $c_{sm}$ denote their Chern–Schwartz–MacPherson classes. Positivity conjecture. All the coefficients appearing in

$$c_{sm}^{\tau_{n,k}^{\circ}},\qquad c_{sm}^{\tau_{n,k}^{\wedge \circ}},\qquad c_{sm}^{\tau_{n,k}^{S \circ}}$$

are non-negative. This was proved for Schubert cells in flag manifolds, but the authors state that they do not know a proof for these determinantal varieties.