This question arose from Homology and Homotopy in the Plane, where it was one of several questions asked (but not answered). I'm posting it separately so I could accept one of the answers there.
Is the following statement true?
In the plane with isolated points removed (possibly infinitely many), two simple (non-self-intersecting) loops are homotopic iff they are homologous.
Here, "loop" means a continuous image of a closed interval, with endpoints at the same point.