Let $(\varphi_{\alpha})_{\alpha\in A}$ be an arbitrary set of $\theta$-psh functions on $X$ and assume that $(\varphi_{\alpha})$ is uniformly bounded from above. If we set $\varphi(x):=\sup_{\alpha\in A}\varphi_{\alpha}(x)$ for each $x\in X$, then the usc regularization $\varphi^{*}$ of $\varphi$ is $\theta$-psh and coincides with $\varphi$ on $X^{\mathrm{qm}}=\bigcup_{\mathcal{X}}\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$.