Equisingularity conjecture for adapted HYM metrics

Let $X$ be a compact Kähler manifold and let $lpha$ be a nef and big class on $X$. Let $T=\theta+dd^c\boldsymbol{\varphi}$ be an adapted closed positive $(1,1)$-current with minimal singularities in $lpha$, where $\theta$ is a smooth representative in $lpha$. Let $E$ be a holomorphic vector bundle with a smooth Hermitian metric $h_0$. Assume that $E$ admits a $T$-adapted HYM metric $h=h_0e^{2S}$ with an $h_0$-Hermitian endomorphism $S$. Equisingularity conjecture. The endomorphism $S$ is equisingular with $\varphi$, that is,

$$|S|_{h_0\otimes h_0^*}+\varphi\in L^{\infty}(X).$$

This conjecture concerns the precise asymptotic behaviour of adapted HYM metrics near the non-Kähler locus of a nef and big class. The preceding discussion explains that the available estimates do not currently ensure equisingularity, so the assertion remains open.