Logarithmic good reduction, monodromy and the rational volume

Let R be a strictly local ring complete for a discrete valuation, with fraction field K and residue field of characteristic p > 0. Let X be a smooth, proper variety over K. Nicaise conjectured that the rational volume of X is equal to the trace of the tame monodromy operator on l- adic cohomology if X is cohomologically tame. He proved this equality if X is a curve. We study his conjecture from the point of view of logarithmic geometry, and prove it for a class of varieties in any dimension: those having logarithmic good reduction.