Hexagon conditional local-move conjecture for punctured-disk resistor networks

Consider a local transformation of a resistor-network subgraph with a central interior boundary vertex, six incident edges with conductances $a,b,c,d,e,f$, a surrounding hexagonal configuration with conductances $g,h,i,j,k,\ell$, and four additional external edges with conductances $m,n,o,p$, together with the corresponding transformed graph with conductances $A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q$. The transformation is the hexagon conditional local move depicted in the source.

Hexagon conditional local-move conjecture. There is a hexagon conditional local move of the displayed form, with the conjectural conductance formulas described in the paper, that preserves the response matrix.

The formulas were checked to preserve every response-matrix entry except the entry in the row and column corresponding to the interior boundary vertex; that computation was too large to complete, although several numerical choices gave equal response matrices. Thus the conjecture remains open.