The conjectured maximal-sink-set formula for Hessenberg cohomology coefficients

Let $h:[n]\to[n]$ be a Hessenberg function, and let $\Gamma_h$ be its associated graph. Let $m=m(\Gamma_h)$ be the number of parts of the partition $\lambda\vdash n$. Let $\mu=(\mu_1,\ldots,\mu_m)$ be a partition of $n-|T|$ satisfying

$$\lambda=(\mu_1+1,\mu_2+1,\ldots,\mu_m+1).$$

For a maximal sink set $T\in\mathrm{SK}_m(\Gamma_h)$, let $\deg(T)$ be its degree and let $c^T_{\mu,j}$ denote the corresponding coefficient for the smaller Hessenberg variety. Maximal-sink-set coefficient formula. For every $i\geq0$,

$$c_{\lambda,i}=\sum_{T\in\mathrm{SK}_m(\Gamma_h)}c^T_{\mu,i-\deg(T)}.$$

The formula is proposed in the paper's section on the general case. The preceding induction proves the analogous non-negativity result only for abelian Hessenberg functions, so this general formula remains conjectural in the source context.