Universal uniform Eberlein compact spaces

A universal space is one that continuously maps onto all others of its own kind and weight. We investigate when a universal Uniform Eberlein compact space exists for weight $\kappa$. If $\kappa = 2^{<\kappa}$, then they exist whereas otherwise, in many cases including $\kappa = \omega _1$, it is consistent that they do not exist. This answers (for many $\kappa$ and consistently for all $\kappa$) a question of Benyamini, Rudin and Wage of 1977.