Suppose that $T$ is a bounded set in $\mathbb{R}^n$ and $f,g$ are two nonnegative functions such that $0\leq f(x)\leq g(x)$ for all $x\geq 0$.

Let $\epsilon_1,\epsilon_2,\dots,$ be a Rademacher sequence. Does it hold that $$ \mathbb{E} \sup_{t\in T} \left| \sum_i \epsilon_i f(|t_i|) \right| \leq C\cdot \mathbb{E} \sup_{t\in T} \left| \sum_i \epsilon_i g(|t_i|) \right| \qquad (*) $$ for some absolute constant $C$?

It looks intuitively true to me but I cannot find it anywhere in the literature. To compare it with the contraction principle that $$ \mathbb{E} \sup_{t\in T} \left| \sum_i \epsilon_i \alpha_i g(|t_i|) \right| \leq \mathbb{E} \sup_{t\in T} \left| \sum_i \epsilon_i g(|t_i|) \right|, \quad |\alpha_i|\leq 1 $$ it seems that the $\alpha_i$ in my case would depend on $t$ and the proof of the contraction principle above does not seem to go through.

Question: Does (*) hold? Is there a reference in the literature or is there a particularly simple proof?