Convergence conjecture for Hermitian–Yang–Mills metrics under divisor degeneration

Let $\overline{E}$ be a holomorphic vector bundle on $\overline{X}$ such that $\overline{E}|_D$ is $c_1(N_D)$-polystable and $\overline{E}$ is $(c_1(D),[\omega_0])$-stable. Let $H_\epsilon$ be the Hermitian–Yang–Mills metric on $\overline{E}$ with respect to the Kähler form $\omega_\epsilon$, and let $H$ be the Hermitian metric constructed in Theorem. Convergence conjecture. There exist smooth functions $f_\epsilon$ on $X$ such that

$$H_\epsilon e^{f_\epsilon}\longrightarrow H \quad\text{in } C^\infty_{\mathrm{loc}}(X).$$

This asks whether the compact Hermitian–Yang–Mills metrics converge locally, after scalar normalization, to the metric constructed on the complete non-compact manifold. The supplied text presents it as a natural problem and gives no resolution, so its status remains open.