Logarithmic aggregation conjecture for fat-shattering dimension

Let $\Omega$ be a domain, let $F_i\subseteq\mathbb{R}^{\Omega}$ for $i\in[k]$, and let $F_{\max}$ be their $k$-fold maximum. Logarithmic aggregation conjecture. For some universal constant $c>0$, for every $\gamma>0$,

$$\operatorname{fat}_{\gamma}(F_{\max})\le c\operatorname{Log}(k)\sum_{j=1}^k\operatorname{fat}_{\gamma}(F_i).$$

This conjecture would give a general logarithmic-in-$k$ upper bound for the fat-shattering dimension of maxima of function classes. The supplied text gives matching results for particular affine classes, but does not state a resolution of this general claim.