Shenfeld–van Handel's hard Lefschetz kernel conjecture for supercritical collections

Let $X$ be a smooth projective variety of dimension $n$. For a collection of nef classes $\mathcal{L}=(L_1,\ldots,L_{n-2})$ on $X$, assume the supercritical condition

$$\operatorname{nd}(L_I)\geq |I|+2\quad\text{for every }I\subset[n-2],$$

where $L_I=\sum_{i\in I}L_i$. Let $\mathbb{L}$ denote the associated hard Lefschetz map, and define

$$V_{\mathcal{L},\operatorname{eff}}=\operatorname{span}_{\mathbb{R}}\{[D]:D\in\operatorname{Prime}(X),\ \mathbb{L}\cdot[D]=0\}.$$

Shenfeld–van Handel's conjecture. One has

$$\ker\mathbb{L}=V_{\mathcal{L},\operatorname{eff}}.$$

In particular, $\mathbb{L}$ is a hard Lefschetz class if and only if $\mathbb{L}\cdot[D]\neq0$ for every prime divisor $D$ on $X$. This conjecture extends the characterization of extremal classes for convex polytopes to arbitrary smooth projective varieties through positivity theory; its resolution is not specified in the source.