Injective edge-function conjecture for length-area transferal

Let $G=(\mathcal{V},\mathcal{E})$ be a simplicial complex. A triangle edge choice function assigns to each triangle of $G$ one of its edges, and it is injective when distinct triangles are assigned distinct edges. Injective edge-function conjecture. Length-area transferal holds for $G$ if and only if there exists an injective triangle edge choice function. The authors know no simplicial complex with an injective edge choice function for which length-area transferal fails; the conjecture therefore proposes that injectivity is sufficient, while the stated equivalence remains open.