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:
$$\left\{(x,\sin\frac{1}{x}):0<x<1\right\}$$
$$0\times[-2,1]$$
$$[0,1]\times -2 $$
$$1\times[-2,\sin 1]$$

![picture of the above union][1]


  [1]: https://i.sstatic.net/cFvJD.jpg