The odd-degree annihilator correction conjecture

Let $V_n$ be an $n$-dimensional vector space, let $f$ be a generic form of odd degree $d>3$ in the exterior algebra, and let $a_{n,d}(t)$, $p_{n,d}(t)$, and $\Delta_{n,d}(t)$ be the Hilbert-series quantities defined in the paper. Put

$$\tau_{n,d}(t)=a_{n,d}(t)-\max\left(p_{n,d}(t),\Delta_{n,d}(t)\right).$$

Odd-degree annihilator correction conjecture. For odd $d>3$,

$$\tau_{n,d}(t)= \begin{cases} t^{v(v-1)/2} & \text{if there exist }v,s\in\mathbb{N},\\ v>0,\\ n-d=-1+\frac52v+\frac12v^2,\\ d=5+2vs,\\\\ 0 & \text{otherwise.} \end{cases}$$

The formula is inferred from computational data for the difference between the actual annihilator Hilbert series and the two predicted lower bounds; it is used to derive formulas for the associated principal-ideal series.