Conjecture on the multiplicative compatibility of the geometric isomorphisms

Let $P$ and $Q$ be the projective modules and let $\psi(P)$ and $\psi(Q)$ be the corresponding subvarieties of the Springer fibre. The graded vector-space isomorphisms identify endomorphism and homomorphism spaces with cohomology groups:

$$\operatorname{End}_{\mathfrak{g}}(P)\cong H^*(\psi(P)),\qquad \operatorname{Hom}_{\mathfrak{g}}(P,Q)\cong H^*(\psi(P)\cap\psi(Q)).$$

Multiplicative compatibility conjecture. The isomorphism $\operatorname{End}_{\mathfrak{g}}(P)\cong H^*(\psi(P))$ is a ring homomorphism, and the isomorphism

$$\operatorname{Hom}_{\mathfrak{g}}(P,Q)\cong H^*(\psi(P)\cap\psi(Q))$$

is compatible with the corresponding left and right module structures over $\operatorname{End}_{\mathfrak{g}}(P)$ and $\operatorname{End}_{\mathfrak{g}}(Q)$, identified with the cohomology rings of $\psi(P)$ and $\psi(Q)$. This would strengthen the known graded-vector-space correspondence by recovering the multiplicative and module structures, but the source does not state a resolution.