The Hilbert-series conjecture for two quadratic forms in an exterior algebra

Let $E$ be an exterior algebra over $\mathbb C$ with $n$ generators, let $f,g\in E$ be generic quadratic forms, and let $S$ be the corresponding commutative polynomial ring with generic linear forms $\ell_1,\ell_2$. For each $s$, let $a(n,s)$ count lattice paths inside the rectangle $(n+2-2s)\times(n+2)$ from the bottom-left to the top-right corner, using only steps $(x,y)\mapsto(x+1,y+1)$ and $(x,y)\mapsto(x-1,y+1)$. Exterior-algebra Hilbert-series conjecture. The Hilbert series of $E/(f,g)$ equals that of

$$S/(x_1^2,\ldots,x_n^2,\ell_1^2,\ell_2^2),$$

and equals $1+a(n,1)t+a(n,2)t^2+\cdots$. This conjecture relates generic quadratic quotients of exterior algebras to power ideals; the source gives it as a concluding open conjecture.