Conjecture on the $h$-vector of the coarse Hall–Littlewood–Schubert series at $Y=-1$

Let $r=\binom{n+1}{2}$, and define the coefficients $h_{n,0}^-,\ldots,h_{n,r-1}^-\in\mathbb{N}_0$ by

$$\mathsf{HLS}_n(-1,X)=\frac{\sum_{i=0}^{r-1}h_{n,i}^-X^i}{(1-X)^r}.$$

The $Y=-1$ $h$-vector conjecture. The coefficients satisfy

$$\sum_{i=0}^{r-1}h_{n,i}^- = \frac{\binom{n}{2}!}{\prod_{i=1}^{n-1}(2i-1)^{n-i}},$$

and $h_{n,1}^-=2^n-1-\binom{n+1}{2}$. Moreover, $h_{n,i}^-=0$ if and only if $(n,i)=(2,1)$. The value at $Y=-1$ is notable because the source observes that each tableau contribution $\Phi_T(-1)$ is either zero or a power of $2$, but the source gives no resolution of the conjectured coefficient properties.