Point-to-point distance in first passage percolation on (tree) x Z

We consider first passage percolation (FPP) on T_d x Z, where T_d is the d-regular tree (d>=3). It is shown that for a fixed vertex v in the tree, the fluctuation of the distance in the FPP metric between the points (v,0) and (v,n) is of the order of at most log n. We conjecture that the real fluctuations are of order 1 and explain why.