Quadrangulation conjecture

Let rooted maps and rooted quadrangulations be counted by generating series $M(u,x,y,z)$ and $Q(u,x,y,z)$, where $u,x,y,z$ mark genus, faces, vertices, and edges, respectively. Their generating series satisfy

$$2 Q(u^2,x,y,z) = M(4u^2,y+u,y,xz^2) + M(4u^2,y-u,y,xz^2).$$

Quadrangulation conjecture. There is an explicit bijection preserving genus and number of edges between rooted quadrangulations and rooted maps with certain decorations that proves this identity.

The conjecture asks for a bijective explanation of the generating-series identity relating rooted maps and quadrangulations. The supplied text states that such a bijection exists, but gives no further resolution evidence.