Closure of the inverse image under the projection map

Question

Let $S$ be a subsemigroup of a semitopological semigroup $(T,+)$, let $e$ be an idempotent in $T\setminus S$ such that $e\in cl_T(S)$, let $\mathcal{E}$ be a subsemigroup of $S\times S$ such that $(e,e)\in cl_{T\times T}(\mathcal{E})$, and let $\pi_1:T\times T\rightarrow T$ be the projection homomorphism onto the first coordinate.

Question: For $A\subseteq \pi_1[\mathcal{E}]$, if $e\in cl_T(A)$, then can we say that $(e,e)\in cl_{T\times T}((A\times S)\cap\mathcal{E})$?

Definitions:

1. $e$ is an idempotent if $e+e=e$.
2. $cl_T(S)$ is the topological closure of $S$ in $T$.

Thanks in advance for any help or suggestion.