The admissible-lift conjecture for local-global convergence

A local-global convergent sequence is a sequence of finite graphs whose local statistics and local-global statistics converge; a modeling limit is a modeling graph representing the limiting local statistics. A class of graphs is nowhere dense if it is nowhere dense in the standard sparse-graph sense.

Admissible-lift conjecture. Every local-global convergent sequence of graphs in a nowhere dense class has a modeling limit.

The preceding theorem establishes only the existence of a modeling quasi-limit, and the source suggests that refining the notion of admissible lift should yield the reverse direction. Thus the asserted existence of a genuine modeling limit remains open.