The factorization conjecture for numerical ranges of compressed shifts

Let $\mathbb{D}^2$ be the bidisk, let $\theta=\lambda\frac{\widetilde p}{p}$ be a rational inner function on $\mathbb{D}^2$, and let $S_\theta^1$ and $S_\theta^2$ denote the two compressed coordinate shifts associated with $\theta$. A function $\theta$ factors as a product of one-variable finite Blaschke products when

$$\theta=B_1(z_1)B_2(z_2),$$

for finite Blaschke products $B_1$ and $B_2$.

Factorization conjecture. If $\theta$ does not factor in this way, then $W(S_\theta^1)$ and $W(S_\theta^2)$ are open. If $\theta$ does factor in this way, then $W(S_\theta^1)$ and $W(S_\theta^2)$ are closed.

This conjecture proposes that the openness or closedness of the numerical ranges of the two compressed shifts is determined exactly by whether the rational inner function has the indicated product factorization. It is motivated by the degree-$(1,n)$ cases, but the general assertion is presented as speculative and its resolution is not given here.