The weight-gcd conjecture for tightly constrained convex toric domains

Let $X_\Omega$ be a convex toric domain with weight sequence $(c;a_1,\dots,a_s;b_1,\dots,b_t)$, and let $\lambda$ be the $\Omega$-perimeter. A convex toric domain is tightly constrained when its capacity function is eventually strictly increasing, equivalently when every sufficiently large positive integer occurs among its ECH capacities. Let $Y_{\Sigma(\Omega)}$ be the toric variety associated to $\Omega$, and let $D_\Omega$ be its balance divisor.

Weight-gcd conjecture. If

$$\operatorname{gcd}\{c,a_1,\dots,a_s,b_1,\dots,b_t\}=1,$$

then $\Omega$ is tightly constrained.

This conjecture would characterize tight constraint by a simple arithmetic condition on the weight sequence. The paper notes that tight constraint is known when at least one weight equals $1$; the general equivalence with the gcd condition remains open.