Complete-graph-cycle bridge-sum orbit-length conjecture

Let K(v_i){{% \mathchoice {\raisebox{-.1mm}{\includegraphics[height=2ex]{Bridge}}} {\raisebox{-.1mm}{\includegraphics[height=2ex]{Bridge}}} {\raisebox{-.6mm}{\includegraphics[height=2ex]{Bridge}}} {\raisebox{-.5mm}{\includegraphics[height=2ex]{Bridge}}} }} C(v_j) be a bridge \sum of a complete graph on $n$ vertices and a cycle graph on $\nu$ vertices, and let $w_\sigma$ be the winding-number parameter determined by the initial labeling $\sigma$. Complete-graph-cycle bridge-\sum conjecture. The orbit length of toric promotion on K(v_i){{% \mathchoice {\raisebox{-.1mm}{\includegraphics[height=2ex]{Bridge}}} {\raisebox{-.1mm}{\includegraphics[height=2ex]{Bridge}}} {\raisebox{-.6mm}{\includegraphics[height=2ex]{Bridge}}} {\raisebox{-.5mm}{\includegraphics[height=2ex]{Bridge}}} }} C(v_j) is

$$w_\sigma(n+\nu-1)(n+\nu).$$

The conjecture proposes a labeling-dependent formula for bridge sums with a cycle component, extending the paper's results for complete-graph bridge sums; it remains open.