The Tosatti–Weinkove conjecture on logarithmic poles in nef classes

Let $X$ be a compact $n$-dimensional complex manifold, and let $\{\beta\}\in H^{1,1}(X,\mathbb{R})$ be nef with $\int_X\beta^n>0$. Fix points $x_1,\ldots,x_N\in X$ and positive real numbers $\tau_1,\ldots,\tau_N$ satisfying

$$\sum_{i=1}^{N}\tau_i^n<\int_X\beta^n.$$

A function is $\beta$-plurisubharmonic if it is a $\beta$-PSH function. The Tosatti–Weinkove conjecture. There exists a $\beta$-PSH function $\varphi$ with logarithmic poles at $x_1,\ldots,x_N$, such that in a coordinate neighbourhood $(z_1,\ldots,z_n)$ centered at $x_i$,

$$\varphi(z)\leq\tau_j\log|z|+O(1),$$

where $|z|^2=|z_1|^2+\cdots+|z_n|^2$. The Kähler version is known by Demailly–Păun, and the conjecture is known in dimensions $2$ and $3$, with partial results in general dimensions and under the assumption that $\beta$ is semi-positive. In the supplied source, the parser marks this candidate as resolved.