Sufficiency: a $T$-witness or an $M$-witness is turned into a labeling by gathering secrets along the first tree or forest, exchanging them (through a hub edge, or around the cycle), and distributing them along the second one. Necessity: contract the maximal sets supporting two edge-disjoint spanning trees, bound the number of calls in the quotient from below by $2r - 4$, split into the cases $2r - 3$ edges (a $T$-witness) and $2r - 4$ edges (canonical form of the call order, a protected extension of a four-cycle to an $M$-witness of the quotient, and a lift to the original graph through a compatible choice of ports).