Nonnegative symmetric decomposition conjecture for fixed-point-free involution Eulerian polynomials

Let $\mathcal{J}_{2n}$ be the set of fixed-point-free involutions in $\mathfrak{S}_{2n}$, and let $J_{2n,k}$ count those with $k$ descents. Define

$$J_{2n}(t)=\sum_{k=1}^{n}J_{2n,k}t^k.$$

Using the symmetric expansion

$$J_{2n}(t)=\sum_{k=1}^{n}b_{2n,k}t^k(1+t)^{2n-2k},$$

Nonnegative decomposition conjecture. For $n\geq 9$ and $k\geq 1$, the coefficients $b_{2n,k}$ are nonnegative integers.

The coefficients are given by explicit alternating sums and are shown to grow with $n$ for each fixed $k>1$, but the stated nonnegativity for all $n\geq 9$ and $k\geq 1$ remains unproved in the source.