Greedy matching conjecture for delooped Khovanov complexes

Let $T$ be a tangle diagram, let $\Psi\llbracket T\rrbracket$ be its delooped Khovanov complex, and let $M_{\operatorname{gr}}$ be the greedy partial matching on the graph $G(\Psi\llbracket T\rrbracket)$.

Greedy matching conjecture. For any tangle diagram $T$, the greedy matching $M_{\operatorname{gr}}$ is a Morse matching on the delooped Khovanov complex $\Psi\llbracket T\rrbracket$.

The construction automatically satisfies the finiteness, matching, and isomorphism conditions of discrete Morse theory; the unresolved issue is the absence of directed cycles. The claim would imply acyclicity of the greedy matching for every tangle diagram.