  We study the question whether copies of $S^1$ in $\mathrm{SU}(3)$ can be
amalgamated in a compact group. This is the simplest instance of a fundamental
open problem in the theory of compact groups raised by George Bergman in 1987.
Considerable computational experiments suggest that the answer is positive in
this case. We obtain a positive answer for a relaxed problem using theoretical
considerations.
