Markov width conjecture for cycles

Let $C$ be a cycle of length $n$, with edges $12, 23, \ldots, n1$. For a vertex $i$, let $d_i$ denote its associated model parameter, and let $\mu(C,d)$ be the Markov width of the cycle model. The indices below are considered cyclically modulo $n$. Markov width conjecture for cycles. The Markov width satisfies

$$\mu(C,d)=\max_{i=1,\ldots,n}\mu(K_3,(d_i,d_{i+1},d_{i+2})).$$

Repeated toric fiber products of cycles reduce computations of Markov width to the three-cycle, which motivates this conjectural reduction; the supplied source does not establish the equality in general.