The $8_5$ rotation-number conjecture for strongly invertible Legendrian knots

Let $\overrightarrow{\mathscr K}$ be a strongly invertible oriented Legendrian knot of type $8_5$, and let $\operatorname{tb}(\overrightarrow{\mathscr K})$ and $\operatorname{rot}(\overrightarrow{\mathscr K})$ denote its Thurston–Bennequin and rotation numbers. The $8_5$ rotation-number conjecture. If

$$\operatorname{tb}(\overrightarrow{\mathscr K})=-11,$$

then

$$\operatorname{rot}(\overrightarrow{\mathscr K})=0.$$

The claim is motivated by the absence of a strongly invertible representative with the same data as a known Legendrian representative having rotation number $2$; it remains open.