Generalized gap threshold conjecture for tangled cakes

Let $\mathcal{T}$ be a tangle, and let $n$ agents have monotone continuous valuations. The Generalized gap threshold conjecture. A tangle $\mathcal{T}$ guarantees connected envy-free allocations for $n$ agents if and only if $n$ is no greater than $\mathcal{T}$'s generalized gap threshold. The generalized gap threshold is the smallest integer $t$ for which $\mathcal{T}$ has a gap $\geq 2$ generalized cutset of cardinality $t$. The conjecture proposes that the obstruction supplied by the generalized gap $\geq 2$ lemma is the only obstruction to envy-free connected allocations for tangled cakes.