In a topological group, for any neighborhood $U$ of the origin, there is another such neighborhood with the property that $V.V\subseteq U.$ I conjecture a similar property for topological inverse semigroups.

Conjecture: Suppose for each idempotent $e$ we have an open set $U_e$ containing it. Then there is an open set $V_e$ such that $e\in V_e\subseteq U_e$ and $V_e. V_f\subseteq V_{ef}$ for all idempotents $e, f$.

Is this seem like a reasonable conjecture? I need this property in a very special case of a result that I am working on. I am looking for proof or related references.