The following seems like a counterexample to the conjecture as originally stated, allowing different size coins. It doesn't seem like the big coin, with diameter $1-\epsilon$, can move right, up, or down. (I apologize for the poor drawing.) (I haven't done "formal" algebra to verify this, but just looking at it, it seems to be so.)

![square with big coin and four small coins][1]

Edit by OP: Here's another look at your idea.

[![enter image description here][2]][2]

  [1]: https://i.sstatic.net/2R92Y.jpg "square with big coin and four small coins"
  [2]: https://i.sstatic.net/AOzRS.png