Suppose, I have randomly assigned individuals to one of two treatment. $T$ is a treatment dummy. My treatment influences the probability that a binary variable $E$ takes the value one. Denote this probability as $P(E=1)$. In turn, this change in $P(E=1)$ impacts the outcome $Y$. Importantly, the outcome also can be directly affected by the treatment. So we have the following causal chains:
Causel chain one: $T-> P(E=1)->Y$
Causel chain two: $T->Y$.
Is there any possibility to identify the direct effect $T->Y$ net of the indirect effect via $P(E=1)$?