The strong Pick set conjecture for rational inner functions on the bidisc

Let $f$ be a rational inner function on $\mathbb D^2$, and let $V=Z_p$ be an irreducible inner variety. Write $\deg_i(f)$ and $\deg_i(p)$ for the degrees in the variable $z_i$. Strong Pick set conjecture. If

$$\deg_i(f)<\deg_i(p)\quad\text{for }i=1,2,$$

then $V$ is a strong Pick set for $f$, whereas if

$$\deg_i(f)\geq\deg_i(p)\quad\text{for }i=1,2,$$

then $V$ is not a strong Pick set for $f$. This conjecture proposes that the one-variable degree criterion for strong Pick sets extends to the bidisc; the source does not report a resolution.