The no-real-roots conjecture for the Stern polynomials B_{h_n}

For $n\in\mathbb{N}$, define

$$h_n=\frac{2}{3}(2^{2n}-1)(2^{2n+1}+1)+1.$$

Let $B_{h_n}(t)$ denote the associated Stern polynomial. No-real-roots conjecture. For each $n\in\mathbb{N}$, the polynomial $B_{h_n}(t)$ is completely complex: the equation

$$B_{h_n}(t)=0$$

has no real roots.

This conjecture strengthens the preceding computational suggestion that the sequence $(h_n)$ supplies further solutions of the main congruence; the source gives no proof or resolution.