Gorenstein conjecture for directed tree Gaussian models

Let $T$ be a directed tree, let $I_T$ be the toric ideal associated with its Gaussian tree model, and let $\mathbb{C}[\Sigma]/I_T$ be the corresponding quotient ring.

Gorenstein tree conjecture. The quotient ring $\mathbb{C}[\Sigma]/I_T$ is Gorenstein if and only if every vertex of $T$ has degree at most three.

The ring is known to be normal and Cohen–Macaulay for every tree, so the conjecture concerns the sharper Gorenstein property. The paper leaves this characterization open.