# Does a compact ANR have a local equiconnecting function which connects distinct points by simple paths?

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

1. For every $$(x,y) \in V$$, the path $$f(x,y,-) \colon [0,1] \rightarrow X$$ starts at $$x$$ and ends at $$y$$.
2. 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.