Extension of Poincaré-dual independence to types E7 and E8

Let $h$ be a Hessenberg function and let $\operatorname{Hess}(N,h)$ be the associated regular nilpotent Hessenberg variety. For a Hessenberg function $h'\subset h$, write $[\operatorname{Hess}(N,h')]$ for its Poincaré dual in $H^*(\operatorname{Hess}(N,h))$. Extension conjecture. The set of Poincaré duals

$$\{[\operatorname{Hess}(N,h')]\in H^*(\operatorname{Hess}(N,h))\mid h'\subset h\}$$

is linearly independent for regular nilpotent Hessenberg varieties of types $E_7$ and $E_8$. The corresponding linear-independence theorem is established in types $A_{n-1},B_n,C_n,D_n,E_6,F_4,G_2$; the assertion for types $E_7$ and $E_8$ is presented as the remaining extension.