Koszulness and confluence conjecture for the family of associative–coassociative properads

Let $E$ be the $\mathbb{S}$-bimodule generated by an associative product and a coassociative coproduct. For $a=(a_1,a_2,a_3,a_4)\in\mathbb{C}^4$, let $\mathcal{P}_a$ be the properad defined by the associative and coassociative relations together with the quadratic relation $\between_a$, and let $\varphi_a:\mathcal{A}\boxtimes\mathcal{C}\to\mathcal{P}_a$ be the induced morphism, where $\mathcal{A}$ is the properad of associative algebras and $\mathcal{C}$ is the properad of coassociative coalgebras.

Conjecture. For every $a\in\mathbb{C}^4$, the following are equivalent: the properad $\mathcal{P}_a$ induces a confluent system; the morphism $\varphi_a$ is a bijection in weight $3$; and $\mathcal{P}_a$ is Koszul.

The conjecture proposes that confluence, the weight-three behavior of the canonical morphism, and Koszulness coincide throughout this family. The supplied text gives no resolution status beyond presenting it as a conjecture.