Local Weak Limits and Unimodularity

Local weak limits of finite graphs are unimodular (involution-invariant) random countable locally-finite rooted graphs. Is the converse true -- does every such infinite graph arise as some local weak limit? Posed in our 2007 paper this has been called the Aldous-Lyons conjecture. See the 2014 Hatami - Lovasz - Szegedy survey Limits of locally-globally convergent graph sequences for discussion of this general topic.