Existence of a competitive online selector for convex body chasing

Let $d4a6$ denote the family of convex sets in $d53d^d$, let $d4a6^*$ be the set of finite strings with alphabet $d4a6$, and let an online selector be a map $S:d4a6^*\tod53d^d$ satisfying $S(K_1,\ldots,K_t)\in K_t$ for every finite sequence of convex sets. Its movement cost on a sequence of convex sets, from a starting state $x_0$, is

$$\operatorname{cost}_S(\mathbf K):=\sum_{t\geq 1}\left\|S(\mathbf K_1,\ldots,\mathbf K_{t-1})-S(\mathbf K_1,\ldots,\mathbf K_t)\right\|,$$

where the norm is Euclidean, and $\operatorname{cost}^*(\mathbf K)$ is the infimum of this quantity over offline selectors. The selector is $\omega$-competitive if $\operatorname{cost}_S(\mathbf K)\leq\omega\operatorname{cost}^*(\mathbf K)$ for every sequence $\mathbf K$. Competitive-selector conjecture. For any $d\in\mathbb N$, there exists an $\omega$-competitive online selector for some $\omega>1$. The conjecture asks for a dimension-dependent constant-competitive deterministic strategy for chasing arbitrary convex bodies in Euclidean space. The source provides no resolution status; the existence of such selectors is the central question in the convex body chasing problem.