I will use the same notation and definitions in High Dimensional Probability, by Roman Vershynin.

I have a sub-Gaussian vector $y$, in $\mathbb{R}^n$ and sub-Gaussian norm $C$ non dependent on $n$. I also assume that $y$ respects the concentration of the norm property, which I define as

$\frac{1}{\sqrt n} (|y|^2 - \mathbb{E}|y|^2)$ is a sub-exponential random variable with sub-exponential norm $C'$ non dependent on $n$, where $|\cdot|$ is the Euclidean norm;

(note that this property holds, for example, for standard Gaussian vectors).

The question is the following: does such a vector respect also the following property?

$\sum_i s_i \,(y_i^2 - \mathbb{E}y_i^2)$ is a sub-exponential random variable with sub-exponential norm $C''$ non dependent on $n$, where $|s| = 1$, and $s_i \geq 0 \; \forall i$.

Note that we are not assuming the independence of the entries of $y$, and that both sub-Gaussianity and concentration of the norm are necessary properties for the thesis to hold. In particular, here is an example of a sub-Gaussian vector that does not respect the concentration of the norm property:

$v = Z g$, where $g$ is a standard Gaussian vector, and $Z$ is a scalar random variable uniform in $[0, 1]$.

However, I'm not sure the result is true in general, since I wasn't able neither to prove it nor to find a counterexample.