Brändén's real-rootedness conjecture for the normalized Boros–Moll polynomials $R_n$

Write the Boros–Moll polynomial as $P_n(x)=\sum_{i=0}^n d_i(n)x^i$. For each integer $n\geq 1$, define

$$R_n(x)=\sum_{i=0}^n\frac{d_i(n)}{(i+2)!}x^i.$$

Brändén's conjecture for $R_n$. For any $n\geq 1$, the polynomial $R_n(x)$ has only real zeros. The paper presents this as the second real-rootedness conjecture related to the Boros–Moll polynomials, and the supplied parser gives no resolution evidence.