Irreducible-component positivity conjecture for non-Archimedean Hodge–Riemann relations

Let $V$ have dimension $n$, let $\rho$ be an irreducible $GL_n({\cal O})$-component of $C^\infty(Gr_{n-i,n}(\mathbb F))$, and let ${\cal D}_i$, $P_i$, and ${\cal L}^{\alpha,-}_\rho$ be as in the analytic reformulation: ${\cal L}^{\alpha,-}_\rho$ is the scalar by which the $GL_n({\cal O})$-equivariant operator $(-1)^i{\cal D}_{n-i}\circ R_{i,n-i}$ acts on $\rho$. Irreducible-component positivity conjecture. If $\rho$ is contained in the orthogonal complement of $\operatorname{Ker}({\cal D}_i)$ in $L^2(Gr_{n-i,n}(\mathbb F))$ and ${\cal D}_i(\rho)\subset P_i$, then

$${\cal L}^{\alpha,-}_\rho>0.$$

By multiplicity-freeness and Schur's lemma, this is equivalent to the analytic Hodge–Riemann positivity conjecture above. The supplied text gives no resolution status.