Conjecture on real-rootedness of the derived Boros–Moll polynomial $Q_n$

For $n\geq 1$, let

$$P_n(x)=\sum_{k=0}^n d_k(n)x^k,$$

where

$$d_k(n)=2^{-2n}\sum_{j=k}^n 2^j{2n-2j\choose n-j}{n+j\choose j}{j\choose k},$$

and define

$$Q_n(x)=\sum_{k=0}^n\bigl(d_k(n)^2-d_{k-1}(n)d_{k+1}(n)\bigr)x^k.$$

Real-rootedness conjecture for $Q_n$. For any $n\geq 1$, the polynomial $Q_n(x)$ has only real zeros. This conjecture is motivated by Brändén's theorem that the operator sending coefficients $a_k$ to $a_k^2-a_{k-1}a_{k+1}$ preserves real-rootedness for real-rooted polynomials with nonnegative coefficients. Its status is not specified in the supplied text.