Fejes Tóth's sausage conjecture for TS-packings of unit balls

Let $N>1$ and $3\leq d\leq 41$. Consider an arbitrary TS-packing of $N$ unit balls in $\mathbb{E}^d$. Fejes Tóth's sausage conjecture. The volume of their convex hull is at least the volume of the convex hull of $N$ non-overlapping unit balls whose centers lie on a line segment of length $2(N-1)$.

This asks whether the linear packing, or sausage, minimizes the convex-hull volume among TS-packings in the remaining dimensions. The claim is stated as already proved for $d=2$ and, via Betke and Henk's result, for all $d\geq 42$; the dimensions $3\leq d\leq 41$ remain under consideration.