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Existence of a knot whose integer λ⁻ exceeds all Rasmussen and torsion bounds

Ascertain whether there exists a knot K such that the integer-coefficient invariant (K) defined via homogeneous, non–quantum-increasing chain maps between the universal Khovanov chain complexes (K; ℤ[G]) and (U; ℤ[G]) satisfies (K) > 1/2 |s_F(K)| and (K) > 𝔲(K; F) for every field F, i.e., exceeds both the Rasmussen-based and torsion-based lower bounds simultaneously for all coefficients.

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Background

The authors exhibit many pairs of knots where their λ− invariant strictly improves both Rasmussen and torsion bounds. They ask whether this phenomenon can occur for the unknot, i.e., whether there exists a single knot K whose λ− to the unknot is strictly larger than both half the Rasmussen invariant (over any field) and the maximal G-torsion order (over any field).

They note that such an example cannot exist over field coefficients, but it may still exist over ℤ, and they are unaware of an algebraic obstruction in the integral setting.

References

The question whether there exists such a pair of knots with $J$ the unknot remains open. More succinctly, is there a knot $K$ with $(K) > \tfrac{1}{2}|s_{\mathbb{F}}(K)|$ and $(K) > \mathfrak{u}(K;\mathbb{F})$ for all fields~$\mathbb{F}$?

— Khovanov homology and refined bounds for Gordian distances (2409.05743 - Lewark et al., 9 Sep 2024) in Section "λ- and Î⁻-distance between torus knots" (following Example exa:lambda-coefficients-matter)