Shift-strategy characterization conjecture for the symmetric group

Let $\mathbb{S}_n$ be the symmetric group on $n$ elements. A shift strategy is the strategy described in the paper in which the message classes are determined by the cyclic shifts, with Alice indicating a shift agreeing with the input permutation at the maximum number of positions. An optimal strategy is one attaining the maximum success probability.

Shift-strategy characterization conjecture. Every optimal strategy is a shift strategy.

This conjecture is a more restrictive structural characterization than the Latin-strategy conjecture. The paper presents it as an open question; no proof or counterexample is supplied.