The boundedness conjecture for intrinsic volumes

Let $n\in\mathbb{N}$, $D>0$, and $\kappa\in\mathbb{R}$. For a closed smooth connected Riemannian manifold $(M^n,g)$, write $V_i(M)$ for its intrinsic volumes. The boundedness conjecture. There exists a constant $C(n,D,\kappa)$ such that if $M$ has diameter at most $D$ and sectional curvature at least $\kappa$, then

$$|V_i(M)|\leq C(n,D,\kappa)$$

for every $i=0,1,\dots,n$. This is motivated by known bounds for volume, Euler characteristic, and the integral of scalar curvature; the general estimate for all intrinsic volumes is conjectural.