  Let $A=\mathcal{A}(E,\sigma)$ be a $3$-dimensional quantum polynomial algebra
where $E$ is $\mathbb{P}^{2}$ or a cubic divisor in $\mathbb{P}^{2}$, and
$\sigma\in \mathrm{Aut}_{k}E$. Artin-Tate-Van den Bergh proved that $A$ is
finite over its center if and only if the order $|\sigma|$ of $\sigma$ is
finite. As a categorical analogy of their result, the author and Mori showed
that the following conditions are equivalent; (1)
$|\nu^{\ast}\sigma^{3}|<\infty$, where $\nu$ is the Nakayama automorphism of
$A$. (2) The norm $\|\sigma\|$ of $\sigma$ is finite. (3) The quantum
projective plane $\mathsf{Proj}_{{\rm nc}}A$ is finite over its center. In this
paper, we will prove for Type S' algebra $A$ that the following conditions are
equivalent; (1) $\mathsf{Proj}_{{\rm nc}}A$ is finite over its center. (2) The
Beilinson algebra $\nabla A$ of $A$ is $2$-representation tame. (3) The
isomorphism classes of simple $2$-regular modules over $\nabla A$ are
parametrized by $\mathbb{P}^{2}$.
