Asai–Iyama's g-tameness and E-tameness conjecture for Jacobi-finite QPs

Let $(Q,W)$ be a non-degenerate Jacobi-finite quiver with potential, and let $\mathcal{J}(Q,W)$ denote its Jacobian algebra. The algebra is called $\operatorname{g}$-tame or $E$-tame according to the corresponding tameness conditions.

Asai–Iyama's conjecture. If $(Q,W)$ is a non-degenerate Jacobi-finite quiver with potential, then

$$\mathcal{J}(Q,W)\text{ is }\operatorname{g}\text{-tame}\quad\Longleftrightarrow\quad\mathcal{J}(Q,W)\text{ is }E\text{-tame}.$$

This conjecture was posed by Asai and Iyama for arbitrary finite-dimensional algebras. The paper studies the equivalence in substantial classes of Jacobian algebras, but the stated general conjecture remains open.