Existence conjecture for distributional limits of uniform spherical triangulations

Let $M\in[6,\infty]$, and let $T^M_j$ be chosen randomly uniformly among isomorphism classes of spherical triangulations with $j$ vertices and maximum degree at most $M$. Given $T^M_j$, choose $o_j$ uniformly among the vertices of $T^M_j$. A distributional limit is a random rooted graph whose finite rooted-neighborhood probabilities are the limits of those for $(T^M_j,o_j)$. Existence conjecture. The distributional limit of $(T^M_j,o_j)$ exists. Existence of this limit is posed in the concluding remarks as an unresolved question for the stated range of $M$, including $M=\infty$.