The Thurston–Bennequin number conjecture for unknotted minimal embeddings of $K_5$

Let $K_5$ be the complete graph on five vertices, and consider an unknotted minimal embedding of $K_5$ as a Legendrian graph. For each $5$-cycle in the embedding, let $tb$ denote its Thurston–Bennequin number. Thurston–Bennequin number conjecture. Any unknotted minimal embedding of $K_5$ will contain at least one $5$-cycle with $tb=-3$, and will not contain a $5$-cycle with $tb=-4$.