The coefficient formula conjecture for generalized Hessenberg varieties

Let $n$ be a positive integer, let $h_2$ be the generalized Hessenberg function under consideration, and write the equivariant Poincaré polynomial of $X_{h_2}$ as

$$\mathcal F(h_2)=\sum_{\lambda\vdash n}c_\lambda\mathsf h_\lambda.$$

For $\lambda=(n-j,j)$, the coefficients $c_\lambda$ are given as follows. If $j<\frac n2$, then

$$q[2]_q^{n-2}\Big([n-j-1]_q[j]_q+[n-j]_q[j-1]_q\Big)+d_{(n-j,j)}[n-j]_q[j]_q,$$

where

$$d_{(n-j,j)}=-2q[2]_q^{n-3}-2q^2[2]_q^{n-5}-\cdots-2q^{j-1}[2]_q^{n-2j+1}-q^j[2]_q^{n-2j-1}+\max(n-2j-2,0)q^{j+1}[2]_q^{n-2j-3}.$$

If $j=\frac n2$, then

$$q[2]_q^{n-2}\left[\frac n2-1\right]_q\left[\frac n2\right]_q+d_{(\frac n2,\frac n2)}\left[\frac n2\right]_q^2,$$

where

$$d_{(\frac n2,\frac n2)}=-q[2]_q^{n-3}-q^2[2]_q^{n-5}-\cdots-q^{\frac n2-1}[2]_q.$$

Coefficient formula conjecture. For every partition $\lambda=(n-j,j)$, the coefficient $c_\lambda$ in the equivariant Poincaré polynomial of $X_{h_2}$ is given by the corresponding formula above. The formula has been verified by computer calculation for $n\leq 13$ using Algorithm 1. The notation $[r]_q$ denotes the $q$-integer and $\mathsf h_\lambda$ the relevant symmetric-function basis element.