See [LV] Loday–Vallette, Algebraic Operads, Section 8.6 for all the definitions.

Let $\mathtt{P} = \mathbb{F}(V)/(R)$ and $\mathtt{Q} = \mathbb{F}(W)/(S)$ be two quadratic operads presented by generators and relations, and let $$\lambda : W \circ_{(1)} V \to V \circ_{(1)} W$$ be a rewriting rule. Let $\mathtt{P} \vee_\lambda \mathtt{Q}$ be the resulting operad, a quotient of $\mathbb{F}(V \oplus W)$.

By Theorem 8.6.5 in [LV], it is sufficient for the projection $\mathtt{P} \circ \mathtt{Q} \to \mathtt{P} \vee_\lambda \mathtt{Q}$ to be injective when restricted to operations of total weight $3$ in generating operations for the rewriting rule to induce a distributive law (basically, the underlying symmetric sequence of $\mathtt{P} \vee_\lambda \mathtt{Q}$ is isomorphic to $\mathtt{P} \circ \mathtt{Q}$).

Some elements of $W \circ_{(1)} V \circ_{(1)} V$, namely those of the form $$q(1,\dots,1,p,1,\dots,1,p',1\dots,1),$$ are roughly speaking "critical monomials" and can be rewritten in two ways, inducing relations in $\mathtt{P} \vee_\lambda \mathtt{Q}$. If $p$ is injective when restricted to weight $3$, then all of these are confluent, i.e. the relations already come from $R$ and $S$.

Question. If all these "critical monomials" are confluent, is $p$ injective in weight $3$, hence $\lambda$ induces a distributive law on $\mathtt{P} \circ \mathtt{Q}$?

My intuition, attempts at proofs, and trying on some simple examples, all say yes. However I've not been able to come up with a full proof, nor with a counterexample. The case I'm most interested in has an infinite number of generating operations, so brute force through seems suboptimal.

See [LV] Loday–Vallette, Algebraic Operads, Section 8.6 for all the definitions.

Let $\mathtt{P} = \mathbb{F}(V)/(R)$ and $\mathtt{Q} = \mathbb{F}(W)/(S)$ be two quadratic operads presented by generators and relations, and let $$\lambda : W \circ_{(1)} V \to V \circ_{(1)} W$$ be a rewriting rule. Let $\mathtt{P} \vee_\lambda \mathtt{Q}$ be the resulting operad, a quotient of $\mathbb{F}(V \oplus W)$.

By Theorem 8.6.5 in [LV], it is sufficient for the projection $\mathtt{P} \circ \mathtt{Q} \to \mathtt{P} \vee_\lambda \mathtt{Q}$ to be injective when restricted to operations of total weight $3$ in generating operations for the rewriting rule to induce a distributive law (basically, the underlying symmetric sequence of $\mathtt{P} \vee_\lambda \mathtt{Q}$ is isomorphic to $\mathtt{P} \circ \mathtt{Q}$).

Some elements of $W \circ_{(1)} V \circ_{(1)} V$, namely those of the form $$q(1,\dots,1,p,1,\dots,1,p',1\dots,1),$$ $$\text{and } q(1, \dots, p(1, \dots, p', \dots, 1), \dots, 1)$$ are roughly speaking "critical monomials" and can be rewritten in two ways, inducing relations in $\mathtt{P} \vee_\lambda \mathtt{Q}$. If $p$ is injective when restricted to weight $3$, then all of these are confluent, i.e. the relations already come from $R$ and $S$.

Question. If all these "critical monomials" are confluent, is $p$ injective in weight $3$? (And hence $\lambda$ induces a distributive law on $\mathtt{P} \circ \mathtt{Q}$ by Theorem 8.6.5 above?)

My intuition, attempts at proofs, and trying on some simple examples, all say yes. However I've not been able to come up with a full proof, nor with a counterexample. The case I'm most interested in has an infinite number of generating operations, so brute force seems suboptimal.