The supercritical collection conjecture for hard Lefschetz classes

Let $X$ be a smooth projective variety of dimension $n$, and let $\mathfrak{L}=(L_1,\ldots,L_{n-2})$ be a supercritical collection of nef classes on $X$. Define

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

Supercritical collection conjecture. One has

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

In particular, $\mathfrak{L}$ is a hard Lefschetz class if and only if $\mathfrak{L}\cdot[D]\neq 0$ for every prime divisor $D$ on $X$. This conjectural description extends the toric characterization of hard Lefschetz classes to smooth projective varieties in the supercritical case; the surrounding discussion presents it as an expectation motivated by known toric results, and no resolution is supplied here.