Discontinuity conjecture for the random billiard walk variance

Let $p\in(0,1)$ be fixed, let $b\in S^{d-1}$ be a direction, and let $\sigma_{b,p}$ denote the standard deviation associated with the random billiard walk. A direction is called rational when it is rational in the sense used for the billiard cutting sequence. Discontinuity conjecture. For fixed $p$, the map $b\mapsto\sigma_{b,p}$ is discontinuous at rational directions. The paper presents this as a conjectural obstruction to approximating an irrational direction by rational directions in a coupling argument; no resolution is given.