For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, assuming $\mu$ is supported on elements whose entries lie in a number field, and the support generates a dense subgroup. It is my understanding that such a result is conjectured to hold for any compact Lie group, and importantly, that $\mu$ need not be supported on matrices with entries in $\overline{\mathbb{Q}}$.
As far as I can tell, this conjecture is wide open. I am searching for references, either papers or expository work, that treat the "non-algebraic" case in the above context, or in related contexts (it need not be a full treatment).
I've heard vague remarks to the effect of "the theory of heights may be useful" or "we only know how to do the initial step in Bourgain-Gamburd-like theorems, without a Diophantine condition, by using number theory." I understand the connotations of the above remarks, but I'm searching for work or exposition that has been done in this direction.
