Jarosz's renorming conjecture for prescribed isometry groups

Let $G$ be a group and let $X$ be a real Banach space with $\dim X \geqslant |G|$. An equivalent renorming of $X$ is a norm equivalent to its original norm, and $\operatorname{Isom}(X)$ denotes the group of linear isometries of the renormed space. Jarosz's conjecture. $X$ admits an equivalent renorming such that

$$\operatorname{Isom}(X)\cong\{-1,1\}\times G.$$

The conjecture was verified for finite groups $G$ and separable spaces $X$, but the source states that it fails for infinite $G$ in general.