The type B hypersimplex reciprocity conjecture

For $n\geq 3$, let $\Psi_{B_n}(t)$ denote the Ehrhart $h^*$-polynomial of the type $B_n$ hypersimplex, and let $E_{D_n}(t)$ denote the type $D_n$ Eulerian polynomial. Type B reciprocity conjecture.

$$\Psi_{B_n}(t)+t^n\Psi_{B_n}(t^{-1})=2E_{D_n}(t).$$

The identity is motivated by computational evidence and proposes a relationship between the type $B$ hypersimplex $h^*$-polynomial and the type $D$ Eulerian polynomial; its general validity remains open.