Multiplicity conjecture for Schubert varieties at arbitrary points

Let $w,\tau\in W/W_{P_d}$. Let $R^+$ and $R_{P_d}^+$ denote the positive roots of the ambient root system and of the parabolic subsystem, respectively. Define $S'_{w,\tau}$ to be the unisets of $(R^+\setminus R_{P_d}^+)^{\ast}$ satisfying the condition that, for every chain of commuting reflections $s_{\alpha_1}>\cdots>s_{\alpha_t}$ with $s_{\alpha_i}\in S'_{w,\tau}$, one has $w\geq\tau s_{\alpha_1}\cdots s_{\alpha_t}$. Let $M$ be the maximum cardinality of an element of $S'_{w,\tau}$. Multiplicity conjecture. The multiplicity of $X(w)$ at $\tau$ is

$$\operatorname{mult}_{\tau}X(w)=|\\{S\in S'_{w,\tau}:|S|=M\\}|.$$

This conjecture gives a combinatorial formula for the multiplicity of a Schubert variety at an arbitrary point, using maximal unisets satisfying the Bruhat-order condition. Its status is unresolved in the supplied source.