Geodesic connectivity criterion for Cegrell class E

Let $\Omega$ be the domain under consideration, let $\mathcal{E}$ be the Cegrell class of negative plurisubharmonic functions on $\Omega$, and let $g_u$ denote the Green–Poisson residual function of $u$. A pair of functions $u_0,u_1\in\mathcal{E}$ is geodesically connected if there exists a plurisubharmonic geodesic joining them. Geodesic connectivity conjecture. For $u_0,u_1\in\mathcal{E}$, the functions $u_0$ and $u_1$ can be connected by a plurisubharmonic geodesic if and only if

$$g_{u_0}=g_{u_1}.$$

This would characterize geodesic connectivity by equality of the leading singularity data. The paper explains that the result would also imply the idempotency property $g_{g_u}=g_u$ for all $u\in\mathcal{PSH}^-(\Omega)$, but the conjecture itself is presented as unresolved.