A family of algebras that connects positroids, the Theta conjecture, the Delta conjecture, and Eulerian quasisymmetric functions

We define a family of algebras $\mathbb{Q}[\mathbf{x}_n]/I_{n,a,b}$ whose Hilbert series, when $n\le a+b\le n+2$, $a>0$, and $0<b<n$, enumerate certain positroid cells by codimension. Equivalently, these series enumerate certain affine permutations by length. We determine the graded $\mathfrak{S}_n$