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Existence of a knot with λ strictly stronger than both s-invariants and torsion bounds for all fields

Determine whether there exists a knot K such that the invariant (K) = (K, U; Z) strictly exceeds both (i) one-half of the absolute value of every Rasmussen invariant s_F(K) over all fields F and (ii) the maximal local G-torsion order 𝔲(K; F) over all fields F; equivalently, construct such a knot or prove that none exists.

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Background

The authors exhibit many pairs of knots K, J for which their invariant (K, J) is strictly larger than both the Rasmussen-based lower bounds and the torsion-based lower bounds over all fields, highlighting the enhanced strength of their construction.

They then ask whether an analogous strict improvement can be realized in the one-sided case against the unknot, i.e., whether there exists a single knot K for which the knot-to-unknot version (K) dominates both half of |s_F(K)| and the torsion bound uniformly over all fields.

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 Remark near the end of Subsection “λ- and λ^−-distance between torus knots” (following Example exa:lambda-coefficients-matter)