Kotrbatý's Hodge–Riemann conjecture for valuations

Let $W$ be a finite-dimensional real vector space of dimension $n$, and let $V_i$, $i=0,1,\dots,n$, denote the intrinsic volumes. For $i\leq n/2$, a valuation $\phi\in Val_i^\infty(W)$ is primitive if $\phi\cdot (V_1)^{n-2i+1}=0$, and a valuation $\psi\in Val_{n-i}^\infty(W)$ is co-primitive if $\psi\ast (V_{n-1})^{\ast(n-2i+1)}=0$. Define Hermitian forms

$$Q(\phi):=(-1)^i\phi\cdot\bar\phi\cdot (V_1)^{n-2i}$$

and

$$\tilde Q(\psi)=(-1)^i\psi\ast\bar\psi\ast V_{n-1}^{\ast(n-2i)}.$$

Kotrbatý's Hodge–Riemann conjecture. Let $i\leq n/2$. If $\phi\in Val_i^\infty(W)$ is a non-zero primitive even valuation, then $Q(\phi)>0$; if it is a non-zero primitive odd valuation, then $Q(\phi)<0$. If $\psi\in Val_{n-i}^\infty(W)$ is a non-zero co-primitive valuation, then $\tilde Q(\psi)>0$. This is the valuation-theoretic analogue of the Hodge–Riemann bilinear relations from Kähler geometry; the source presents it as Kotrbatý's conjecture, and no resolution is given here.