Conjecture on ultra-log-concave intrinsic-volume sequences

Let $(a_k)_{k\geq 0}$ be a sequence of positive numbers with $a_0=1$. An infinite-dimensional GB-set is a set for which the intrinsic volumes $\{V_k(K)\}_{k\geq 0}$ form such a sequence.

Conjecture on ultra-log-concave intrinsic-volume sequences. The sequence $(a_k)_{k\geq 0}$ is the intrinsic-volume sequence of some infinite-dimensional GB-set if and only if

$$a_k^2\geq\frac{k+1}{k}a_{k-1}a_{k+1},\qquad k=1,2,\dots.$$

Thus, ultra-log-concavity is conjectured to be not only necessary but also sufficient for an infinite-dimensional intrinsic-volume sequence. The source presents this as an open analogue of the Blaschke problem in Hilbert space; no resolution is supplied.