Injectivity conjecture for the endomorphism algebra of divided power 1-morphisms

Let $I$ be the vertex set of the Dynkin diagram, let $i\in I$, and let $n\geq 1$. Define the graded $\Bbbk$-algebra $H_i^n$ with generators $y_k,z_k$ for $k=1,\dots,n$ and $t_l$ for $1\leq l\leq n-1$, of degrees $2,2,0$, respectively, subject to the symmetric-group, symmetric-algebra, exterior-algebra, and mixed relations given in the construction above. Let $P_i^n$ be the corresponding divided-power 1-morphism in $\mathcal{H}^\Gamma$. The natural map of $\Bbbk$-algebras

$$H_i^n \longrightarrow \operatorname{End}_{\mathcal{H}^\Gamma}(P_i^n)$$

Injectivity conjecture. The natural map is injective. This conjecture identifies the explicitly presented algebra $H_i^n$ with a subalgebra of the endomorphism algebra; the source gives no resolution, and the issue is the faithfulness of the hollow-dot generators in addition to the already understood crossing and solid-dot subalgebra.