Grid-move conjecture for minimal grid diagrams of $(p,2)$ torus knots

Let $T_{p,2}$ be a torus knot, let $K_L$ be a Legendrian knot of topological type $T_{p,2}$, and let $\alpha(T_{p,2})$ denote the grid number of $T_{p,2}$. Consider a grid diagram for $K_L$ of size $\alpha(T_{p,2})$. Grid-move conjecture. There is a sequence of commutation and cyclic permutation moves taking $K_L$ to the standard grid diagram representing it. The conjecture has been verified by a computer program up to $p=7$; commutation and cyclic permutation are standard moves on grid diagrams, with commutation allowed for adjacent non-interleaved rows or columns.