The $m(10_{125})$ strongly invertible Legendrian representative conjecture

Let $\mu(\overrightarrow{\mathscr K})$ denote the Legendrian mirror of an oriented Legendrian knot $\overrightarrow{\mathscr K}$, and let $-\mu(\mathscr K)$ denote its reversed Legendrian mirror. The $m(10_{125})$ strongly invertible representative conjecture. Let $\mathscr K$ be a Legendrian knot representing $m(10_{125})$ with $\operatorname{tb}(\mathscr K)=-6$. Then $\mathscr K$ is Legendrian isotopic to $-\mu(\mathscr K)$ but not to a strongly invertible Legendrian knot. This conjecture proposes an obstruction to strong invertibility beyond failure to be isotopic to the reversed Legendrian mirror.