Conjecture on common roots of Ramanujan-type polynomials and roots of unity

Let $p$ be a prime number and $k$ a natural number. Consider the Ramanujan-type polynomial $\mathfrak{R}_{2k+1,p}(z)$ and the polynomial $(pz)^{2k}-1$.

Common-root conjecture. If $2\mid k$, then the only common roots of $\mathfrak{R}_{2k+1,p}(z)$ and $(pz)^{2k}-1$ are $z=\pm i/p$; if $2\nmid k$, then the two polynomials have no common roots. This conjecture describes precisely which zeros of the Ramanujan-type polynomial can also be $2k$th roots of unity after rescaling. The supplied text gives no resolution, so the assertion remains open.