The perimeter bound for NS-families of translates

Let $F4D0\mathcal{F}$ be an NS-family of $n\geq 1$ translates of an origin-symmetric convex domain $F4D0\mathbf{M}$. Then

$$\operatorname{per}_M(\operatorname{conv}(\mathcal{F})) \leq 4n-4+\operatorname{per}_M(\mathbf{M}).$$

This bound generalizes the preceding successive-perimeter result to NS-families, with perimeter measured in the norm whose unit ball is $F4D0\mathbf{M}$. The supplied context presents the claim in a conjecture environment but does not establish whether it is proved or open.