Absolute convex hull of rank 1-correlation matrices?

Does there exist a ''universal'' constant, $$c > 0$$ say, such that for any(!) $$k \in \mathbb{N}$$ every(!) $$k \times k$$-correlation matrix $$\Sigma$$ can be written as $$\Sigma = c\Theta$$, where $$\Theta$$ is an element of the absolute convex hull of $$k \times k$$-correlation matrices of $$\underline{rank \hspace{0.1cm} 1}$$?

I guess there exists a counterexample. However, I haven't found one so far. Any hints and links to suitable references are highly welcome!