Type B real-rootedness conjecture for the polynomials $B_n$

Let $B_n$ be the type B polynomials defined in the paper, for $n\geq 1$. A polynomial is real-rooted if all of its roots are real. Type B real-rootedness conjecture. For all $n\geq 1$, the polynomials $B_n$ are real-rooted. The preceding corollary establishes that the polynomials $B_n$ are $\gamma$-positive, and real-rootedness would further imply $\gamma$-positivity and ultra log-concavity for palindromic polynomials with nonnegative coefficients; the source gives no resolution status for this type B analogue.