Codimension-one reduction conjecture for ESFL contact sub-connections

Let a contact sub-connection admit an ESFL reduction by a partial contact curve of codimension $s>1$. Codimension-one reduction conjecture. The contact sub-connection must also admit an ESFL reduction by a partial contact curve of codimension $1$. If true, this would reduce the classification of ESFL partial contact curve reductions to the single-input case; the source gives the BC system as an example where the conjecture holds, but no general proof.