The extremal ML threshold relation for graphical models

Let $G$ be an undirected graph. Write ${\rm eMLT}(G)$ for its extremal maximum likelihood threshold and ${\rm MLT}(G)$ for its maximum likelihood threshold.

Extremal ML threshold relation. For any undirected graph $G$,

$${\rm eMLT}(G) = {\rm MLT}(G)-1.$$

This relation is motivated by the inequality cited in the supplied context and the computation for the 4-cycle, where ${\rm eMLT}(C_4)=2$; its general validity remains open in the supplied text.