The ribbon cobordism converse to cubiquitous embeddings

Let $L_1$ and $L_2$ be non-split alternating links, let $\Sigma(L_i)$ denote the double cover of $S^3$ branched along $L_i$, and let $\Lambda(L_i)$ denote their black Tait lattices. A stabilization of a lattice is its direct sum with a Euclidean lattice, and a sublattice is cubiquitous in a stabilization when every unit cube based at a lattice point of the stabilization contains a point of the sublattice.

Ribbon cobordism converse. If $\Lambda(L_2)$ admits a cubiquitous embedding into a stabilization of $\Lambda(L_1)$, then there exists a ribbon cobordism from $\Sigma(L_1)$ to $\Sigma(L_2)$.

The paper proves the forward implication for quasi-ribbon cobordisms and consequently obtains monotonicity of the normalized determinant. The converse is described as a very optimistic strengthening of the cubiquity conjecture and remains open.