Convergent matrix-integral decomposition conjecture

Let $H_N$ be the space of Hermitian $N\times N$ matrices, let $V$ be a potential such that the matrix integral below is convergent, and let $C_{ij}$ be the quadratic-form coefficients. Define

$$Z_{\rm conv}=\int_{H_N^p}dM_1\,dM_2\,\dots\,dM_p\,\exp\left\{-{N\over t}\left(\sum_{i,j=1}^p{C_{ij}\over2}\operatorname{Tr}(M_iM_j)-NV(M_1,\dots,M_p)\right)\right\}.$$

A convergent formal solution of the loop equations is a formal solution whose expansion in $N^{-2}$ is convergent.

Convergent matrix-integral decomposition conjecture. The convergent matrix integral $Z_{\rm conv}$ is a finite linear combination of convergent formal solutions of the loop equations: there exist constants $c_i$ and convergent formal solutions $Z_i$ such that

$$Z_{\rm conv}=\sum_i c_iZ_i,\qquad \ln Z_i=-\sum_{g=0}^{\infty}N^{2-2g}F_i^{(g)}.$$

This conjecture concerns the relationship between convergent and formal solutions of the loop equations. The surrounding discussion emphasizes that loop equations can have multiple solutions and that the supplied text gives no resolution of this claim.