Guo–Zeng's gamma-positivity conjecture for involution descent polynomials

Let $\mathcal{I}_n$ be the set of involutions in $\mathfrak{S}_n$, and define

$$I_n(t)=\sum_{\pi\in\mathcal{I}_n}t^{\operatorname{des}(\pi)}.$$

Guo–Zeng conjecture. For every $n\in\mathbb{N}$, there exist coefficients $a_{n,i}\in\mathbb{N}$ such that

$$I_n(t)=\sum_{i=0}^{\lfloor (n-1)/2\rfloor}a_{n,i}t^i(1+t)^{n-1-2i},$$

where $0\leq i\leq\lfloor (n-1)/2\rfloor$.

This is a gamma-positivity expansion for the descent polynomial of involutions. The source presents it as an open problem after noting that unimodality and symmetry were already known.