The irreducibility conjecture for Q_n^{(k)} and V_n

Let $Q_n^{(k)}(x)$ and $V_n(x)$ be the polynomials defined in the paper, with integers $n\geq 1$ and $0\leq k\leq n-1$. Irreducibility conjecture. For all integers $n\geq 1$ and $0\leq k\leq n-1$, the polynomials $Q_n^{(k)}(x)$ and $V_n(x)$ are irreducible over $\mathbb Q$.

The paper proves some partial irreducibility results, including irreducibility of $V_n(x)$ when $2n+1$ is prime, while computations indicate that the stated assertion holds in the full range.