Extension from singular hypersurfaces for plurisubharmonic weights

Let $W$ be the hypersurface and let $\varphi$ be a plurisubharmonic weight for which the upper density $D^+_{\varphi}(W)$ is defined. The theorem asserting the relevant Bargmann–Fock extension property for $W$ is assumed in the source as Theorem. The extension conjecture. Theorem holds for any plurisubharmonic weight $\varphi$ such that

$$D^+_{\varphi}(W)<1.$$

The source proposes this because the proof in the general, possibly singular case uses a stronger curvature hypothesis, while no reason is known why the result should fail for merely plurisubharmonic weights. Whether the extension theorem holds under this weaker hypothesis is left open.