Lalín–Rogers-type conjecture for Ramanujan-type polynomial zeros

For a prime number $p$ and a natural number $k$, define the Ramanujan-type polynomial

$$\mathfrak{R}_{2k+1,p}(z):=\sum_{j=1}^{k}(p^{2j}-1)(p^{2k+2-2j}-1)\frac{B_{2j}}{(2j)!}\frac{B_{2k+2-2j}}{(2k+2-2j)!}(pz)^{2k+2-2j}.$$

Ramanujan-type polynomial zero conjecture. The polynomial $\mathfrak{R}_{2k+1,p}(z)$ has only the real zero $z=0$ of multiplicity $2$; all remaining zeros are non-real, lie on the circle $|z|=1/p$, and are simple. For $p=2$, the corresponding assertion was proved by Lalín and Rogers, while the conjecture proposes the analogous statement for every prime $p$.