Bounded-carrier Brownian box-ball invariance conjecture

Fix $K>0$. Let $W^K$ be the stationary Markov process obtained by conditioning the carrier process associated with Brownian motion with drift $c>0$ to remain below $K$, let $L^K$ be its local time at $0$ with $L^K_0=0$, and set

$$S^K=L^K-W^K+W^K_0.$$

Thus $S^K$ is interpreted as Brownian motion with drift $c>0$ conditioned to stay within $K$ of its past maximum, and let $T$ denote the box-ball transformation on paths.

Bounded-carrier Brownian invariance conjecture. If $S^K$ is the Brownian motion with drift $c>0$ conditioned to stay within $K$ of its past maximum, then

$$TS^K\buildrel{d}\over{=}S^K.$$

This is a conjectured non-periodic analogue of the preceding invariance result. The paper describes a potential proof strategy via stationary Markov processes and scaling limits, but does not establish the claim.