It is known that if $X$ is a (metric) ANR, then $X$ is locally equiconnected, that is, there is a neighborhood $V$ of the diagonal $\Delta X \subseteq X \times X$ and a continuous function $$f \colon V \times [0,1] \rightarrow X$$ such that
- For every $(x,y) \in V$, the path $f(x,y,-) \colon [0,1] \rightarrow X$ starts at $x$ and ends at $y$.
- For every $x \in X$, the path $f(x,x,-) \colon [0,1] \rightarrow X$ is the constant path at $x$.
[Side note: Local equiconnectivity is equivalent to the diagonal map $\Delta \colon X \rightarrow X \times X$ being a Hurewicz cofibration.]
Let us also assume that $X$ is compact. My question is: Can we choose the $V$ and $f$ such that when $x \neq y$ in the 1st condition, the path connecting them is a simple path?
Remark: It follows from Lemma 2.1 of the paper "A remark on simple path fields in polyhedra of characteristic zero" by Fadell that the answer is yes when $X$ is a finite simplicial complex. I am interested in a (strict) generalization of this result.
