Conjectures on real roots and reducibility of the polynomials B_{p_{k,n}}

For $k\in\mathbb{N}_{\geq2}$ and $n\in\mathbb{N}_{+}$, define

$$p_{k,n}=2^{2n+k}-3\cdot2^{n+k-1}+2^k-3$$

and

$$N_k(n)=\left|\{t\in\mathbb{R}:B_{p_{k,n}}(t)=0\}\right|.$$

Conjectures for $B_{p_{k,n}}$. (1) If $k\equiv0\pmod 2$, there is $c_k\in\mathbb{N}_{+}$ such that $B_{p_{k,n}}(t)$ has exactly one real root for $n\in\mathbb{N}_{\geq c_k}$; under the same assumptions, $y=B_{p_{k,n}}(t)$ is increasing for $t\in\mathbb{R}$. (2) If $k\equiv1\pmod2$, there is a number $C_k$ such that $N_k(n)\leq C_k$ for each $n\in\mathbb{N}_{+}$. (3) $B_{p_{k,n}}(t)$ is reducible if and only if $k\in\mathbb{N}_{\geq3}$ and $n=k-1$; in that case

$$B_{p_{k,k-1}}(t)=\left(1+2t\frac{t^{k-2}-1}{t-1}\right)\left(B_{2,k-2}(t)+2t^{k-2}(1+t)\right),$$

and both factors are irreducible in $\mathbb{Q}[t]$.

These claims are based on experiments with $2\leq k\leq10$ and $n\leq10^3$. The source notes that Eisenstein's criterion proves irreducibility for the two factors in the stated special case, but gives no resolution of the full assertions.