mathematics – Are infinite loops possible in the game Sublimergy?

I’d like to know if infinitely repeating sequences of moves (i.e. cycles) are possible in the following game:

Sublimergy is a two-player game played on the spaces (cells) of an
initially empty hexagonal grid of hexagons (board).

Two stones, or a stone and an empty cell, see each other if they lie
on the same straight line of adjacent cells and there are no other
stones between them along that line.

You control a cell if the number of stones of your color seen by that
cell is greater than half the number of cells (empty or occupied)
adjacent to that cell.

On your turn, place a stone of your color on an empty cell not
controlled by your opponent or remove an enemy stone from a cell you
control. The last player to move wins.

Note that it is possible to keep material constant on an infinite board with this configuration:

 x x x x x x x x x x x x x …
x o o o o o o . . . . . . .…
 x . x . x . x . x . x . x …

X keeps removing O’s pieces from the left while O extends the o line to the right.