The Hadamard-closure conjecture for the subalgebra of a Q-polynomial graph

Let $\Gamma=(X,\mathcal R)$ be a finite, undirected, connected graph without loops or multiple edges, with diameter $D\geq 1$, and let $A$ be its adjacency matrix. An ordering $\{V_i\}_{i=0}^d$ of the eigenspaces of $A$ is Q-polynomial if

$$\sum_{\ell=0}^i V_\ell=\sum_{\ell=0}^i (V_1)^{\circ \ell}\qquad (0\leq i\leq d).$$

The graph $\Gamma$ is Q-polynomial if it has at least one such ordering. Let $M$ be the subalgebra of $\operatorname{Mat}_X(\mathbb R)$ generated by $A$. Hadamard-closure conjecture. If $\Gamma$ is Q-polynomial, then

$$B\circ C\in M$$

for all $B,C\in M$. This would extend a central closure property of the subconstituent algebra from distance-regular graphs to the broader class of Q-polynomial graphs, whose distance-regularity is not assumed here; the conjecture's status is not determined in the source.