Sharp One-Dimensional Sub-Gaussian Comparison in Convex Order

We prove that any random variable $X$ whose moment generating function is point-wise upper bounded by that of $G\sim \mathcal{N}(0,1)$ must be dominated by $G\mathrm{/}\mathbb{E}[\mathrm{\mid }G\mathrm{\mid }]$ in convex order, meaning $\mathbb{E}[f(X)]\le \mathbb{E}[f(G\mathrm{/}\mathbb{E}[\mathrm{\mid }G\mathrm{\mid }])]$ for all convex $f$. This is sharp as witnessed by $X\sim \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}(\{-1,1\})$ and $f(x)=\mathrm{\mid }x\mathrm{\mid }$.