The exceptional Thurston–Bennequin conjecture for the knot $9_{42}$

For a strongly invertible Legendrian knot, let its Thurston–Bennequin number be denoted by $\operatorname{tb}$, let $\overline{\operatorname{tb}}(K)$ be the maximal Thurston–Bennequin number among Legendrian representatives of a knot type $K$, and let $\overline{\operatorname{tb}}_e(K)$ be the corresponding maximum restricted to strongly invertible Legendrian representatives. The $9_{42}$ Thurston–Bennequin conjecture.

$$\overline{\operatorname{tb}}_e(9_{42})<\overline{\operatorname{tb}}(9_{42}).$$

This is based on experiments showing that the maximal known representative of $9_{42}$ is not strongly invertible; the claim remains an experimental conjecture.