Equivariant positivity conjecture for Chern-Mather classes of symmetric-subgroup orbit closures

Let $G$ be a reductive algebraic group with involution $\theta$, let $K=G^{\theta}_0$ be the identity component of the fixed-point group, and let $B\supset H$ be a $\theta$-stable Borel subgroup and maximal torus of $G$. Set $T=(H\cap K)_0$, choose the positive system of roots whose positive root spaces lie in $\operatorname{Lie} B$, let $W$ be the Weyl group of $G$, and write $X=G/B$. For $w\in W$, let $Y_w=\overline{BwB}/B\subset X$; the classes $[Y_w]_T$ form a basis of $H^T_*(X)$ over $H^*_T$. For a $K$-orbit closure $Y\subset X$, let $c_M^T(Y)\in H^T_*(X)$ denote its $T$-equivariant Chern-Mather class. Equivariant positivity conjecture. When $c_M^T(Y)$ is expressed in the Schubert basis $[Y_w]_T$, each coefficient is a sum of monomials in the positive simple roots of $G$, restricted to $T$, with nonnegative integer coefficients. These Chern-Mather classes generalize the equivariant fundamental classes whose positivity is known, and the conjecture is stated in the broader setting of symmetric subgroups; the paper verifies it in an example, while the general assertion remains open.