Smooth regularization conjecture for plurisubharmonic functions on compact Kähler spaces

Let $(X,\omega)$ be a compact Kähler space, and let $\varphi$ be an $\omega$-psh function on $X$. Smooth regularization conjecture. The function $\varphi$ can be written as the pointwise limit of a decreasing sequence $(\varphi_i)$ of smooth $\omega$-psh functions. This is the natural extension to singular compact Kähler spaces of the known Richberg-type regularization result on compact Kähler manifolds; whether it holds in the stated singular setting remains open.