Here's a counterexample, a hexagon that falls into a nontrivial 2-cycle:

The coordinates of one of the hexagons is approximately:

{{-2.0951, 1.04954}, {-1.86954, -0.931927}, {-0.480952, -2.1886}, {1.92123, -1.76496}, {2.57605, 1.13906}, {-0.0516838, 2.69689}}

For those who want to experiment more, here's a link to the Mathematica notebook I used to generate it.

edit: Jeremy Rickard points out that (with points interpreted as complex numbers) this example is (up to scaling and rotation) a sum of eigenvectors $z_\omega=(1,\omega,\omega^2,\dots,\omega^5)$ with $\omega=e^{2\pi i/6}$ and $z_{\omega^2}=(1,\omega^2,\omega^4,\dots,\omega^4)$. Indeed, I found that (after rotation by -0.416 radians and a permutation of the coordinates) the above hexagon is $z_\omega+(0.142)z_{\omega^2}$.

I have also updated the above notebook with the code I used to make the animated images that I've edited into his answer.

Here's a counterexample, a hexagon that falls into a nontrivial 2-cycle:

The coordinates of one of the hexagons is approximately:

{{-2.0951, 1.04954}, {-1.86954, -0.931927}, {-0.480952, -2.1886}, {1.92123, -1.76496}, {2.57605, 1.13906}, {-0.0516838, 2.69689}}

For those who want to experiment more, here's a link to the Mathematica notebook I used to generate it.

edit: Jeremy Rickard points out that (with points interpreted as complex numbers) this example is (up to scaling and rotation) a sum of eigenvectors $z_\omega=(1,\omega,\omega^2,\dots,\omega^5)$ with $\omega=e^{2\pi i/6}$ and $z_{\omega^2}=(1,\omega^2,\omega^4,\dots,\omega^4)$. Indeed, I found that (after rotation by -0.416 radians and a permutation of the coordinates) the above hexagon is $z_\omega+(0.142)z_{\omega^2}$.

I have also updated the above notebook with the code I used to make the animated images that I've edited into his answer.

Here's a counterexample, a hexagon that falls into a nontrivial 2-cycle:

The coordinates of one of the hexagons is approximately:

{{-2.0951, 1.04954}, {-1.86954, -0.931927}, {-0.480952, -2.1886}, {1.92123, -1.76496}, {2.57605, 1.13906}, {-0.0516838, 2.69689}}

For those who want to experiment more, here's a link to the Mathematica notebook I used to generate it:

Here's a counterexample, a hexagon that falls into a nontrivial 2-cycle:

The coordinates of one of the hexagons is approximately:

{{-2.0951, 1.04954}, {-1.86954, -0.931927}, {-0.480952, -2.1886}, {1.92123, -1.76496}, {2.57605, 1.13906}, {-0.0516838, 2.69689}}

For those who want to experiment more, here's a link to the Mathematica notebook I used to generate it.

edit: Jeremy Rickard points out that (with points interpreted as complex numbers) this example is (up to scaling and rotation) a sum of eigenvectors $z_\omega=(1,\omega,\omega^2,\dots,\omega^5)$ with $\omega=e^{2\pi i/6}$ and $z_{\omega^2}=(1,\omega^2,\omega^4,\dots,\omega^4)$. Indeed, I found that (after rotation by -0.416 radians and a permutation of the coordinates) the above hexagon is $z_\omega+(0.142)z_{\omega^2}$.

I have also updated the above notebook with the code I used to make the animated images that I've edited into his answer.