The asymptotic Carathéodory-rank conjecture for normal polytopes

For a lattice polytope $P$, define its Carathéodory rank $\operatorname{CR}(P)$ to be the smallest natural number $k$ such that, for every natural number $c$ and every lattice point $z\in cP$, there exist lattice points $x_1,\ldots,x_k\in P$ and integers $a_1,\ldots,a_k\ge0$ with

$$z=a_1x_1+\cdots+a_kx_k,\qquad a_1+\cdots+a_k=c.$$

For $d\in\mathbb{N}$, let

$$\operatorname{CR}(d)=\max\{\operatorname{CR}(P)\mid P\subset\mathbb{R}^d\text{ is a normal polytope}\}.$$

Asymptotic Carathéodory-rank conjecture. The ratio $\operatorname{CR}(d)/d$ is monotonically converging to $2$ as $d\to\infty$. The source records a counterexample to the integral Carathéodory property giving $\limsup_{d\to\infty}\operatorname{CR}(d)/d\ge7/6$, but the supplied status evidence says the conjecture is disproved.