I think this version of the topologist's sine curve is compact and simply connected but there are obvious neighborhoods with no simply connected open refinement.
Take the union of the following: $$\left\{(x,\sin\frac{1}{x}):0<x<1\right\}$$ $$0\times[-2,1]$$ $$[0,1]\times -2 $$ $$1\times[-2,\sin 1]$$
