The converse $L^2$-optimality conjecture for strongly upper semi-continuous functions

Let $D\subset\mathbb{C}^n$ be a domain, and let $\varphi$ be a strongly upper semi-continuous function on $D$, meaning that for every $x\in D$ and every subset $S\subset D$ of measure zero,

$$\limsup_{\substack{z\in D\setminus S \\ z\to x}}\varphi(z)=\varphi(x).$$

Here, $\varphi$ is $L^2$-optimal if, for every Stein coordinate domain $U\subset D$, every smooth strictly plurisubharmonic function $\phi$ on $U$, and every Kähler metric $\omega$ on $U$, the equation $\bar\partial u=f$ can be solved for every $\bar\partial$-closed $(n,1)$-form $f\in L^2_{(n,1)}(U;\operatorname{loc})$ whenever the right-hand side below is finite, with

$$\int_U |u|^2_{\omega}e^{-\varphi-\phi}\,dV_{\omega}\leqslant\int_U\langle B_{\omega,\phi}^{-1}f,f\rangle_{\omega}e^{-\varphi-\phi}\,dV_{\omega},$$

where $B_{\omega,\phi}=[i\partial\bar\partial\phi,\Lambda_{\omega}]$. The converse $L^2$-optimality conjecture. If $\varphi$ is $L^2$-optimal, then $\varphi$ is plurisubharmonic.