Multiplicity-one generation criterion for blow-ups of projective toric surfaces

Let $\mathbb P=\mathbb P_\Sigma$ be a projective toric surface with Cox ring $\mathbb C[x_1,\dots,x_n]$ and lattice ideal $I_{\mathbb P}$. Let $e$ denote the identity of the torus of $\mathbb P$, and write $\operatorname{Bl}_e\mathbb P$ for its blow-up at $e$. Multiplicity-one generation conjecture. The following are equivalent: (1) the Cox ring of $\operatorname{Bl}_e\mathbb P$ is generated in multiplicity $1$; (2) for every three indices $i,j,k\in\{1,\dots,n\}$,

$$I_{\mathbb P}\not\subseteq \langle x_i,x_j,x_k\rangle^2.$$

The theorem immediately preceding this conjecture proves that condition (1) implies condition (2); the converse is proposed based on the examples checked in the paper and remains open in the stated generality.