Bóna's real-rootedness conjecture for descent polynomials of stack-sortable permutations

Let $W_{t}(n,k)$ be the number of $t$-stack sortable permutations in $\mathfrak{S}_n$ with $k$ descents, and define the descent polynomial

$$W_{n,t}(x)=\sum_{k=0}^{n-1}W_{t}(n,k)x^{k}.$$

Bóna's real-rootedness conjecture. The descent polynomial $W_{n,t}(x)$ has only real zeros for any integer $1 \le t \le n-1$.

Bóna had previously shown that, for fixed $n$ and $t$, $W_{n,t}(x)$ is symmetric and unimodal. The abstract states that Brändén proved this conjecture as a consequence of a more general result, and the paper gives another proof using $\mathbf{s}$-Eulerian polynomials.