Cusp-existence conjecture for the t-split two-periodic Aztec diamond

Let $\alpha$ and $\beta$ be parameters satisfying

$$0<\beta,\alpha<1.$$

Let $t$ be the interface parameter, and let a smooth region mean a smooth region of the limit shape to the left of the interface. Cusp-existence conjecture. A smooth region exists to the left of the interface if and only if

$$\frac{\beta^2}{1+\beta^2}<t.$$

The conjecture makes precise the claim that the existence of the cusp depends only on $\beta$ and $t$, although its location depends on $\alpha$; it identifies the threshold with the cusp position for the corresponding two-periodic model relative to the interface. The source gives no resolution status.