Hyatt's interlacing conjectures for half Eulerian polynomials

Let ${\mathfrak B}_n$ be the Coxeter group of type $B$, realized as signed permutations, and let ${\mathfrak D}_n$ be its subgroup of even signed permutations. Define the half Eulerian polynomials $B_n^+(x)$ and $D_n^+(x)$ by summing $x^{\operatorname{des}_B\sigma}$ and $x^{\operatorname{des}_D\sigma}$, respectively, over elements with positive final entry. For real-rooted polynomials, write $g(x)\preceq f(x)$ when $g$ interlaces $f$.

Hyatt's conjectures. For $n\geq 1$, $B_n^+(x)$ interlaces $x^nB_n^+(1/x)$, and hence $B_n(x)=B_n^+(x)+x^nB_n^+(1/x)$ has only real zeros. For $n\geq 2$, $D_n^+(x)$ interlaces $x^nD_n^+(1/x)$, and hence $D_n(x)=D_n^+(x)+x^nD_n^+(1/x)$ has only real zeros.

The source states that Hyatt confirmed these conjectures in a new version of the cited work, and the paper gives an affirmative answer by proving them.