Bezdek's intrinsic-volume conjecture for ball bodies with fixed inradius

Let $d>1$, $r>0$, and $0<k\leq d-2$. An $r$-ball body is an intersection of finitely many balls of radius $r$ in $\mathbb{E}^d$; its inradius is the radius of the largest ball contained in it. An $r$-lens is the intersection of two balls of radius $r$. Bezdek's intrinsic-volume conjecture. Among $r$-ball bodies of a given inradius in $\mathbb{E}^d$, the $r$-lens is the only one whose $k$-th intrinsic volume is maximal. The surface-area and volume cases discussed immediately beforehand support this conjecture, while the stated range $0<k\leq d-2$ remains conjectural.