Zero-free Taylor polynomial conjecture for square roots of truncated geometric series

Let $d>2$ and define the coefficients $b_k^{(d)}$ by

$$\sqrt{1+z+\cdots+z^d}=\sum_{k=0}^\infty b_k^{(d)}z^k.$$

For each $n\geq 0$, the corresponding partial sum is the polynomial $\sum_{k=0}^n b_k^{(d)}z^k$. Zero-free Taylor polynomial conjecture. For $d>2$, every such partial sum has no zeros in $\overline{\mathbb D}$, where $\mathbb D$ is the open unit disk. This conjecture would extend the zero-free result established in the paper for $d=2$ and is relevant to sharp bounds for segments of bounded power series via the Landau–Szász method.