Suppose some circular coins (not necessarily the same size) are in a frame. The coins may be immobile, as in this example:

On the other hand, they may be free to move, as in these examples (in which the coins can move simultaneously):

It is rather tedious to show algebraically that the coins can move, so I tried to find some general principles that allow us to simply look at diagrams like these and know whether the coins can move.

Conjecture: If circular coins (not necessarily the same size) are in a convex polygonal frame, with each coin touching exactly one edge, then all the coins can move.

Is my conjecture true?

**Remarks about my conjecture**

- The frame must be a polygon, otherwise there would be a counter-example: two coins in the region bounded by $y=x^2-1$ and $y=1-x^2$, as shown below.
- The frame must be convex, otherwise there would be a counter-example, as shown below.
- Every coin must touch an edge, otherwise there would be a counter-example, as shown below.

**EDIT**

Zach Teitler has given a counter-example. I have proposed a second conjecture that avoids this counter-example.

**EDIT2**

My second conjecture also has a counter-example. I have asked another question asking for general principles that are useful in determining whether coins can move.