Limit-shape conjecture for recurrent configurations of well-behaved Bulgarian solitaires

Let a sequence of well-behaved $csigma_n$-solitaires have stable configurations with limit shape $cphi$. Limit-shape conjecture. If $cphi$ is a limit shape of the stable configurations of a sequence of well-behaved $csigma_n$, then $cphi$ is also a limit shape of the recurrent configurations. This conjecture predicts that recurrent configurations remain sufficiently close to stable configurations in the scaling limit. It is left open in the source.