Conjectures on roots, irreducibility and monotonicity of the polynomials B_{s_{i,n}}

Let $s_{i,n}$, for $i\in\{0,1,2,3\}$, be the sequences defined earlier in the paper, and let $B_{s_{i,n}}(t)$ be their associated Stern polynomials. Conjectures for $B_{s_{i,n}}$. (1) If $i=0,2,3$, then $B_{s_{i,n}}(t)$ has exactly one real root for $n\in\mathbb{N}_{\geq2}$. (2) If $i=0,2,3$, then $B_{s_{i,n}}(t)/(t+1)$ is irreducible in $\mathbb{Q}[t]$. (3) $B_{s_{1,n}}(t)$ has exactly three real roots for $n\in\mathbb{N}_{\geq3}$. (4) The factors of $B_{s_{1,n}}(t)$ displayed immediately before the conjecture are irreducible. (5) The function $y=B_{s_{0,n}}(t)$ is increasing for $t\in\mathbb{R}$.

The claims follow computationally motivated factorization and real-root observations; the source provides no proof or resolution.