Geometric recursion for Masur-Veech volumes?

The Masur-Veech volume $b_{g,n}$ of  $\overline{\mathcal{M}}_{g,n}$ can be defined in two ways. Firstly, as the integral over  $\overline{\mathcal{M}}_{g,n}$ of a certain analytic function $B_{g,n}$. Secondly as the coefficient of the leading term of some enumerative problem (here a certain type of quadrangulations in genus $g$). There is a formula for the Masur-Veech volume $b_{g,n}$ as a polynomial of integrals of psi-classes (on various $\overline{\mathcal{M}}_{g',n'}$). This formula can be derived from any of the two definitions (that was respectively done by Mirzakhani and Delecroix-Goujard-Zograf-Zorich). It is likely that the Masur-Veech volumes $b_{g,n}$ satisfy TR and that the functions $B_{g,n}$ satisfies GR. However it is not (yet) transparent from the formulas.