Let $f : \mathbb{R}^k \rightarrow \mathbb{R}^{d_1}$ and let us be given some $n$ points in $\mathbb{R}^k$ as $\{ x_1,..,x_n \}$. Let $T : \mathbb{R}^{d_1} \rightarrow \mathbb{R}^{d_2}$ be linear transformations whose $(1,q)$-norms will be assumed to be bounded by some $r$. Now consider the following quantity where the distribution of $\epsilon$ is uniform,
$$\mathbb{E}_{\epsilon \sim \{ -1,1\}^n}\left[ \sup_{ ||T||_{1,q} \leq r} \left\lVert \sum_{i=1}^n \epsilon_i T [ f(x_i) ] \right\rVert_1 \right] = \mathbb{E}_{\epsilon \sim \{ -1,1\}^n}\left[ \sup_{ ||T||_{1,q} \leq r} \left\lVert T \left[\sum_{i=1}^n \epsilon_i f(x_i) \right] \right\rVert _1 \right].$$
Now does the following upper bound on the above make sense?
$$ \mathbb{E}_{\epsilon \sim \{ -1,1\}^n}\left[ \sup_{ ||T||_{1,q} \leq r} \left\Vert T \left[\sum_{i=1}^n \epsilon_i f(x_i) \right] \right\Vert _1 \right] \leq r \mathbb{E}_{\epsilon \sim \{ -1,1\}^n}\left[ \left\Vert \sum_{i=1}^n \epsilon_i f(x_i) \right\Vert _{\infty} \right] $$
If the above is not true then what is the closest analogue to such an upper bound that might be true?