Convergence of uniform measures on finite sphere triangulations

Let $\mathcal{D}$ be trivial, and let $\nu_k$ be the uniform probability measure on sphere triangulations with at most $k$ faces. The measures are regarded as transverse measures, so convergence means that $\nu_k(\hat{G}_0^A)$ converges for every decorated discrete finite patch $A$. Convergence conjecture. The sequence of measures $\nu_k$ in this construction converges. The paper notes that convergence is not known; a convergent subsequence exists by diagonalization, and its limit is denoted by $\nu_u$.