Surjectivity isomorphism conjecture for endomorphisms of the identity

Let $B$ be the graded algebra under consideration, let $B_\psi=B/\langle b-\psi(b)\mid b\in B\rangle$, and let $C$ be the graded vector space with basis $v_d$ for $d\in\widetilde{\mathbb N}$. The algebra homomorphism

$$S(B_\psi\otimes_\mathbb{F}C)\twoheadrightarrow \operatorname{END}_{\mathcal{H}'_B}\mathbf{1},\qquad \beta\otimes v_d\mapsto c_{\beta,d}$$

is surjective. Endomorphism isomorphism conjecture. The map above is an isomorphism. This would give a complete presentation of the endomorphism algebra of the identity object in the Heisenberg category, strengthening the established surjectivity result.