Characterization conjecture for ESFL reductions by codimension 1 partial contact curves

Let $p(z)$ denote the function associated with an ESFL reduction by a codimension $1$ partial contact curve, and let the truncated Euler operator and the higher-order Euler operators be the operators used in the source. ESFL characterization conjecture. The only examples of systems with ESFL reductions by codimension $1$ partial contact curves have $p(z)$ in the kernel of the truncated Euler operator and also satisfy truncated versions of identities relating higher-order Euler operators to the total derivative operator. This is an informal conjecture about necessary conditions characterizing all such examples; the source gives no proof or resolution.