g-tame incidence algebras of finite posets are tame

Let $P$ be an arbitrary finite poset, and let its incidence algebra over the field $\K$ be denoted by $\operatorname{Inc}_\K(P)$. The algebra is tame if its finite-dimensional representations have tame representation type, and it is $\mathbf{g}$-tame in the sense of $\mathbf{g}$-vector fan theory.

Tameness conjecture for g-tame incidence algebras. If $\operatorname{Inc}_\K(P)$ is $\mathbf{g}$-tame, then $\operatorname{Inc}_\K(P)$ is tame.

For finite simply connected posets, the paper proves the corresponding equivalence between $\mathbf{g}$-tameness and tameness. The conjecture extends the implication to arbitrary, possibly multiply connected, finite posets, where the reduction to concealed algebras of wild type may fail.