Free-generation conjecture for the closed-diagram endomorphism algebra

Let $I$ be the vertex set of the Dynkin diagram. For $i\in I$ and $k\in\mathbb{N}$, let $c_k^i$ be a clockwise circle labeled $i$ containing exactly $k$ hollow dots, and let $c_k$ be a clockwise circle containing one solid dot and $k$ hollow dots; the label of $c_k$ is irrelevant. Let $\operatorname{End}_{\mathcal{H}^\Gamma}(\mathrm{id})$ be the algebra of closed diagrams acting as endomorphisms of the identity 1-morphism. Free-generation conjecture. The algebra $\operatorname{End}_{\mathcal{H}^\Gamma}(\mathrm{id})$ is isomorphic to the symmetric algebra freely generated by

$$\\{c_k^i\\}_{k\geq 0,\,i\in I}\quad\text{and}\quad\\{c_k\\}_{k\geq 0}.$$

This conjecture would give an explicit polynomial description of the closed-diagram algebra and, together with the preceding factorization conjecture, reduce the endomorphism problem for $P_i^n$ to these circle generators. The source gives no evidence of a resolution.