The strong uniform time conjecture for Rubik's cube corners

Fix an arbitrary order on the positions of the corners of a Rubik's cube. Let $X_t$ denote the configuration of the cube at time $t$, and call two corners unlinked when they are unlinked along each of the $X$, $Y$, and $Z$ axes. Strong uniform time conjecture. When every pair of corners of $X_t$ is unlinked, any two corners are equally likely to be placed one before the other or vice versa. Unfortunately, this conjecture does not hold, so the proposed strong uniform time property is refuted.