Positivity conjecture for trigonometric polynomials from square-root coefficients

For $d\geq 2$, define the coefficients $b_k^{(d)}$ by

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

For each $n\geq 0$, define the trigonometric polynomial

$$\mathcal T_n^{(d)}(t)=\sum_{k=0}^n b_k^{(d)}\cos kt.$$

Positivity conjecture. For $d\geq 2$, the trigonometric polynomials $\mathcal T_n^{(d)}(t)$ are positive for every $t\in\mathbb R$. Positivity of these polynomials is suggested as an approach to the zero-free Taylor polynomial conjecture and concerns the coefficients of the non-hypergeometric power series arising from $\sqrt{(1-z^{d+1})/(1-z)}$.