Sturmian maximizing measures as extremal barycentres

For integers $1\le p<q$, let $F_{0,p,q}$ be the multi-valued dynamical system on $S^1$ appearing in the paper, and let a Sturmian measure mean a probability measure whose support is contained in some $1/q$-circle. Sturmian maximizing-measures conjecture. The set of barycentres of probability measures on $S^1$ invariant under $F_{0,p,q}$ has the property that a measure has barycentre on the boundary if and only if it is Sturmian. This predicts that the extremal barycentres are exactly those arising from measures supported in a $1/q$-circle; the supplied excerpt gives no resolution status.