Cohomological conjecture for smooth Hessenberg varieties

Let $(G,K)=(SL(N),SO(N))$ with $N$ odd, and let $\operatorname{Hess}(K/P,{\mathfrak g}_1,\Sigma)\to{\mathfrak g}_1$ be a family of Hessenberg varieties associated to a parabolic subgroup $P\subset K$ and a $P$-invariant subspace $\Sigma\subset{\mathfrak g}_1$. Cohomological conjecture for smooth Hessenberg varieties. The cohomology of smooth Hessenberg varieties can be expressed in terms of Hodge classes coming from the cohomology of partial flag varieties and the cohomology of hyperelliptic curves. This would relate the geometry of Hessenberg varieties to the cohomology of partial flag varieties and hyperelliptic curves; the paper presents the assertion as evidence-supported and does not establish it in general.