Zero-freeness conjecture for optimal polynomial approximants in Hardy spaces

Let $1<p<\infty$, $f\in H^p$, and $n\in\mathbb{N}$. If $f(0)\neq 0$, then the optimal polynomial approximant $q_{n,p}[f,1]$ is the polynomial of degree at most $n$ that best approximates $1/f$ in the relevant $L^p$ setting, and it is zero-free in the closed unit disk $\overline{\mathbb{D}}$. Zero-freeness conjecture. For every such $f$, $q_{n,p}[f,1]$ is zero-free in $\overline{\mathbb{D}}$. This conjecture extends the zero-location phenomena known for least-squares and Shanks-type approximants to the broader range $1<p<\infty$ in Hardy spaces; its general validity remains open.