Dilks–Petersen–Stembridge conjecture on affine Eulerian polynomials

Let $W$ be an irreducible finite Weyl group generated by $\{s_1,s_2,\ldots,s_n\}$, and let $s_0$ be the reflection corresponding to the highest root. For $\sigma\in W$, define its affine descent set by

$$\widetilde{\operatorname{Des}}\,\sigma=\{i\in\operatorname{Des}\,\sigma:1\leq i\leq n\}\cup\{0:\ell(\sigma s_0)>\ell(\sigma)\},$$

and set $\widetilde{\operatorname{des}}\,\sigma=|\widetilde{\operatorname{Des}}\,\sigma|$. Define the affine Eulerian polynomial

$$\widetilde{W}(x)=\sum_{\sigma\in W}x^{\widetilde{\operatorname{des}}\,\sigma}.$$

Dilks–Petersen–Stembridge conjecture. For any irreducible finite Weyl group $W$, the affine Eulerian polynomial $\widetilde{W}(x)$ has only real zeros. The conjecture is an affine analogue of the real-rootedness theorem for descent polynomials of finite Coxeter groups. The affine Eulerian polynomials are known to have unimodal coefficients, but the asserted real-rootedness remains open in the supplied source.