Demailly–Păun-type conjecture for null loci of asymptotically conical Kähler classes

Suppose $X$ is an asymptotically conical Kähler manifold and $[b1] b5 H^{1,1}_{bd}(X,\mathbb{R})$ is a limit of $bd$-almost compactly supported Kähler classes. Then there is a function $c8:X\to\mathbb{R}\cup\{-\infty\}$ such that

$$\alpha+c3qrt{-1}\partial\bar{\partial}\psi geqslant \varepsilon\omega$$

for some asymptotically conical Kähler form $\omega$. Define

$$\operatorname{Null}(\alpha):=\bigcup_{\int_V\alpha^{\dim V}=0}V,$$

where the union is taken over all compact, irreducible, analytic subvarieties $V\subset X$. Null-locus conjecture. Then $\operatorname{Null}(\alpha)$ is an analytic subvariety, and $\psi$ can be chosen so that it is smooth on $X\backslash\operatorname{Null}(\alpha)$ and

$$\{\psi=-\infty\}=\operatorname{Null}(\alpha).$$

This is the proposed asymptotically conical analogue of the compact Kähler result of Demailly–Păun and Collins–Tosatti; establishing the analyticity of the null locus and the existence of a potential with precisely this singular set remains open.