Minimal grid representative conjecture for Legendrian knots

A grid diagram represents a Legendrian knot, with its grid size measuring the complexity of that representative. For a topological knot $K$, let $\overline{tb}(K)$ denote its maximal Thurston--Bennequin number and let $\alpha(K)$ denote its arc index. A Legendrian knot has maximal Thurston--Bennequin number when its Thurston--Bennequin number equals $\overline{tb}(K)$. Minimal grid representative conjecture. Any Legendrian knot of maximal Thurston--Bennequin number has a grid diagram representative of minimal grid number. More generally, for a topological knot $K$, any Legendrian knot of type $K$ and Thurston--Bennequin number $\overline{tb}(K)-m$ can be represented by a grid diagram of size $\alpha(K)+m$. This conjecture underlies the claimed completeness of the atlas: if true, representatives at each prescribed Thurston--Bennequin level can be found among grid diagrams of the stated size. The supplied text gives no resolution status or supporting theorem beyond presenting it as the completeness principle for the algorithm.