Analytic Hodge–Riemann positivity conjecture for non-Archimedean valuations

Let $V$ have dimension $n$, let $Val_i^\infty(V)$ denote the degree-$i$ smooth translation-invariant valuations, and identify valuations represented by functions on the relevant Grassmannian. For $i\leq n/2$, let $P_i\subset Val_i^\infty(V)$ be the primitive subspace

$$P_i=\{\phi\in Val_i^\infty(V)\mid \phi\cdot V_1^{n-2i+1}=0\}.$$

Let ${\cal P}_i$ be the $GL_n({\cal O})$-invariant subspace of $C^\infty(Gr_{n-i,n}(\mathbb F))$ consisting of functions orthogonal to $\operatorname{Ker}({\cal D}_i)$ and satisfying ${\cal D}_i(f)\in P_i$. Analytic Hodge–Riemann conjecture. For every $\hat f\in {\cal P}_i\setminus\{0\}$,

$$(-1)^i\bigl(\hat f,({\cal D}_{n-i}\circ R_{i,n-i})\hat f\bigr)>0.$$

This is the analytic reformulation of the non-mixed Hodge–Riemann relations in degree $i$. The supplied text gives no resolution status for this formulation.