The arbitrary-collection conjecture for hard Lefschetz kernels

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

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

and let

$$V_{\mathfrak{L},\deg}=\operatorname{span}_{\mathbb{R}}\{\mu_*(\widetilde{\alpha}-\widetilde{\beta}):(\widetilde{\alpha},\widetilde{\beta})\text{ is a }\mu^*\mathfrak{L}\text{-degenerate pair on }\widetilde{X}\},$$

where the span is over all modifications $\mu:\widetilde{X}\to X$ and all such pairs on $\widetilde{X}$. Here a pair of nef classes is $\mathfrak{L}$-degenerate when, for a fixed ample class $A$, it satisfies $\mathfrak{L}\cdot\alpha\cdot\beta=0$ and $\mathfrak{L}\cdot A\cdot\alpha=\mathfrak{L}\cdot A\cdot\beta$. Arbitrary-collection conjecture. One has

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

In particular, $\mathfrak{L}$ is a hard Lefschetz class if and only if $V_{\mathfrak{L},\deg}=\{0\}$ and $\mathfrak{L}\cdot[D]\neq0$ for every prime divisor $D$ on $X$. This is the proposed extension of the supercritical picture to arbitrary nef collections; the paper states that the characterization is substantially subtler and requires new inputs, while presenting positive evidence and examples rather than a proof.