A topological space $X$ is semi-locally simply connected if, for any $x\in X$, there exists an open neighbourhood $U$ of $x$ such that any loop in $U$ is homotopically equivalent to a constant one in $X$ or, equivalently, if the functor $$\Pi_1(U)\rightarrow\Pi_1(X)$$ induced by the inclusion $U\subseteq X$ factorizes through a groupoid in which for each pair of objects there is *exactly* one morphism.

My question is: is it true that if a space $X$ is such that, for any open subset $U\subseteq X$ ($X$ included), $U$ is semi-locally simply connected, then $X$ must be

locally simply connected?

Notice that this implies that $X$ is locally path connected. I tried to post this question on stack exchange some months ago, but I didn't receive any answer. It was just something that came to my mind while I was studying for my master thesis.