The differential-square-zero conjecture for the coloured HOMFLY-PT complexes

For a crossing with colours $i$ and $j$, let $d_k^{\pm}$ be the successive differentials in the associated complex, with $0\leq k\leq i-2$. Differential-square-zero conjecture.

$$d_{k+1}^{\pm}d_k^{\pm}=0$$

for all $0\leq k\leq i-2$. Establishing these identities is necessary for the displayed sequences to be complexes; the authors describe a direct verification as difficult and state that they plan to prove the conjecture by another technique.