WLLN for first-passage percolation on finite graphs

Update (6/2015). Solved myself (bad form, I know) in this paper.

My survey paper on "interacting particle systems repackaged as social dynamics" has many open problems. My favorite concerns general conditions for a first passage percolation time on a general graph to be weakly concentrated around its mean (in the WLLN sense) -- see section 7.2 of the survey paper.