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

Let $r>r_0>0$, $N>1$, $d>k>0$, and let $P=\{\mathbf p_1,\dots,\mathbf p_N\}\subset\mathbb{E}^d$ satisfy $r_{cr}(P)=r_0$, where $r_{cr}$ is the circumradius. Let $\operatorname{conv}_r P$ denote the intersection of the radius-$r$ balls centered at the points of $P$, and let $S_{r,r_0,d}$ be the $r$-spindle with circumradius $r_0$. Bezdek's circumradius conjecture.

$$V_k(S_{r,r_0,d})\leq V_k\left(\operatorname{conv}_r P\right).$$

Thus the $r$-spindle is conjectured to minimize the $k$-th intrinsic volume among the indicated $r$-ball bodies with the prescribed circumradius. The $k=d$ case is stated in the source as a theorem; the displayed generalization to $d>k>0$ is presented as a conjecture.