The intrinsic-volume upper bound for intersections of congruent balls

Let ${\mathbb E}^d$ be Euclidean $d$-space, let $P=\{\mathbf{p}_1,\mathbf{p}_2,\dots,\mathbf{p}_N\}\subseteq{\mathbb E}^d$, and let $P^r$ denote the intersection of the closed balls of radius $r$ centered at the points of $P$. For a compact convex set, write $r_{cr}(P)$ for its circumradius, let $V_k$ denote its $k$th intrinsic volume, and let $L_{r,\rho,d}$ be an $r$-lense in ${\mathbb E}^d$ with inradius $\rho$. The intrinsic-volume upper-bound conjecture. If $r>r_0>0$, $N>1$, $d>k>0$, and $r_{cr}(P)=r_0$, then

$$V_k(P^r)\leq V_k\left(L_{r,r-r_0,d}\right).$$

The conjecture extends the established volume bound for intersections of congruent balls to intrinsic volumes. The supplied context does not state whether this conjecture has been resolved.