The classification conjecture for non-orientably fillable torus knots

Let $T(p,2)$ denote the torus knot with relatively prime parameters $p$ and $2$, where $p<0$. A torus knot is non-orientably fillable if it admits a non-orientable Lagrangian filling.

Classification conjecture. The only non-orientably fillable torus knots are of the form $T(p,2)$ with $p<0$.

This conjecture proposes a complete description of non-orientably fillable torus knots, extending the cases established in the paper and excluding all other torus-knot parameters. Its resolution would clarify the limitations of non-orientable Lagrangian fillings beyond the explicitly constructed examples.