Weak Shanks Conjecture on zeros of bidisk optimal polynomial approximants

Let $H^2(\mathbb{D}^2)$ be the Hardy space on the bidisk, and let $f$ be a polynomial in this space with no zeros in the closed bidisk. For each multidegree, an optimal polynomial approximant (OPA) to $1/f$ is a polynomial minimizing the distance from $pf$ to $1$ in the Hardy-space norm. Weak Shanks Conjecture. The OPAs to $1/f$ cannot vanish inside the open bidisk. This refinement excludes the known counterexamples, whose defining functions have zeros in the bidisk, and remains unresolved after almost fifty years.