This is a modification of the somewhat naive question that I asked below.

Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\gamma_1, \gamma_2, ..., \gamma_n))$ an arbitrary element of $\mathbb{R}^n$, suppose that $\|\gamma_1 u_1 + \gamma_2 u_2 + \cdots \gamma_n u_n\| \geq \|\gamma\|_{\ell^\infty}$. Then is there a bound independent of $n$ (but depending on the cotype constant) of

$$\inf_\epsilon \sup_{\beta \geq 0} \frac{\sum \beta_i\cdot \frac{1}{\sqrt{n}}}{\|\sum \beta_i \epsilon_i u_i \|}$$

where $\epsilon = ((\epsilon_1,\epsilon_2, ..., \epsilon_n))$ varies over all sequences of $\pm 1$, and $\beta \geq 0$ means that $\beta_i \geq 0$ for all $i$.

My suspicion is that this is actually false, but I haven't constructed an explicit counterexample.

*The original question I asked was resolved by two answers below. (I had left out a $\sqrt{n}$ term that I meant to include, but even with this, the answer is no.)*

Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\gamma_1, \gamma_2, ..., \gamma_n))$ an arbitrary element of $\mathbb{R}^n$, suppose that $\|\gamma_1 u_1 + \gamma_2 u_2 + \cdots \gamma_n u_n\| \geq \|\gamma\|_{\ell^\infty}$. (It is important that this inequality be strict, not just an 'up to a constant' sort of bound. What is going on becomes more clear if a picture is drawn.) Is there a bound independent of $n$ on

$\sup_\beta \frac{\sum \beta_i}{\|\beta_1 u_1 + \cdots + \beta_n u_n\|}?$

If not, what if we impose the additional condition that the $\beta_i$ all be non-negative? One might guess the bound is linear with the cotype-2 constant.