Motivic standard conjecture of Hodge type

Let $\mathcal X$ be a smooth projective scheme of relative dimension $d$ over a field or over the ring of integers of a $p$-adic field, satisfying the Künneth standard conjecture, and let $\mathcal L$ be a relatively ample line bundle. Let $\mathfrak P^n_{\mathcal L}(\mathcal X)$ be the motivic primitive part, defined as the kernel of

$$\cup[\mathcal L]^{d-n+1}:\mathfrak h^n(\mathcal X)\longrightarrow\mathfrak h^{2d-n+2}(\mathcal X)(d-n+1).$$

Let $\eta_{\mathcal L,\mathrm{mot}}$ be the associated motivic intersection form. Motivic standard conjecture of Hodge type. There is a notion of polarization on motives such that

$$\eta_{\mathcal L,\mathrm{mot}}:\mathfrak P^n_{\mathcal L}(\mathcal X)\otimes\mathfrak P^n_{\mathcal L}(\mathcal X)\longrightarrow\mathds 1(-n)$$

is a polarization for each $\mathcal X$ and each ample divisor $\mathcal L$. This would provide a motivic generalization of the Hodge--Riemann relations and would imply the cycle-theoretic standard conjecture of Hodge type through realization; its general validity remains open.