The universal rho-degeneracy conjecture for knots

Let $K$ be a knot in $S^{3}$. A knot is $\rho$-degenerate when its reduced spectral sequence converges at $\underline{E}_{1}$ and the induced filtration $\underline{\rho}$ is constant on each nontrivial factor $\widehat{HF}(\Sigma(K),\mathfrak{s})$. Universal $\rho$-degeneracy conjecture. Every knot $K\subset S^{3}$ is $\rho$-degenerate.

The conjecture would extend the established $\rho$-degeneracy result for two-bridge knots to all knots. Equivalently, by the preceding observation, it predicts convergence at $\underline{E}_{1}$ together with constancy of the induced filtration on each nontrivial Heegaard Floer homology factor.