Toric regulator graph construction conjecture

Let $P$ be the projection from the toric regulator target to the relevant quotient, let $\operatorname{reg}_t$ denote the toric regulator, and let $\operatorname{reg}_?$ be the graph-theoretic regulator constructed from semistable curves, their dual graphs, harmonic representatives, and rational functions.

Toric regulator graph construction conjecture. The projected toric regulator equals the graph-theoretic regulator:

$$P\operatorname{reg}_t=\operatorname{reg}_?.$$

The statement occurs at the end of a proposed construction and no proof or status information is supplied in the source. The notation and the precise target are specialized to that construction.