Continuous $(m-1)$-psh + $m$-qpsh + vanishing $m$-Hessian measure $\implies m$-psh?

This is a question about the definition and characterization of $m$-subharmonic ($m$-psh) functions in pluripotential theory (complex Hessian equations). Cross posted at https://math.stackexchange.com/questions/5135220/continuous-m-1-psh-m-qpsh-vanishing-m-hessian-measure-implies-m-p Let $\Omega\subset\mathbb{C}^n$ be a domain, and let $\omega=dd^c|z|^2$ be the standard Euclidean Kähler form. Let $w\in C^0(\Omega)$ be a continuous function on $\Omega$ such that: $w$ is $(m-1)$-psh; $w$ is $m$-qpsh (i.e., $w+C|z|^2$ is $m$-psh for some constant $C>0$); The $m$-th complex Hessian measure vanishes: $$ (dd^c w)^m \wedge \omega^{n-m} = 0 $$ as positive Borel measures on $\Omega$. Question: Is $w$ necessarily $m$-psh? Remarks For $C^2$ smooth functions, this is trivially true. For merely continuous functions, the usual tools break down if we only assume $(m-1)$-psh. I suspect an extra condition like $\Delta w\leq C$ (in the distribution sense) might be needed, but I am not certain. Background definitions (standard in the field) The positivity cone for the $\sigma_m$-equation is $$ \Gamma_m = \bigl\{\lambda\in\mathbb{R}^n : \sigma_1(\lambda)>0,\,\dots,\sigma_m(\lambda)>0\bigr\}, $$ where $\sigma_j(\lambda)=\sum_{1\leq i_1<\cdots