The presentation conjecture for regular nilpotent Hessenberg variety cohomology

Let $b[?m\mu=(n)b[0m$ and let $h$ be a Hessenberg function. Write $R$ for the polynomial ring and $J_h$ for the ideal associated with $h$, and let $b[?m\mathfrak{H}(X,h)b[0m$ denote the regular nilpotent Hessenberg variety.

Presentation conjecture. The quotient

$$R/J_h$$

is a presentation for the cohomology ring of $b[?m\mathfrak{H}(X,h)[0m$. Moreover, this presents the cohomology ring with integer coefficients.

This conjecture proposes an algebraic presentation of the cohomology ring of every regular nilpotent Hessenberg variety, extending the preceding correspondence between generators of $R/J_h$ and the Betti numbers. The supplied text does not state whether the conjecture has been proved or remains open.