The abelianized characteristic-algebra topological-knot conjecture

Let $K$ be a Legendrian knot with maximal Thurston–Bennequin number. Let its characteristic algebra be abelianized by imposing $vw=wv$ for all elements $v,w$, and regard the resulting algebra without grading and over $\mathbb{Z}$. Abelianized characteristic-algebra conjecture. The equivalence class of this abelianized characteristic algebra depends only on the topological class of $K$. If true, this would provide a new topological knot invariant; the source describes the claim as based on empirical evidence, and no resolution is given.