We state a conjecture on the repetition threshold of rich sequences over alphabet of any size. It is known to hold for binary and ternary alphabets. We provide two main contributions that may be helpful for the proof on larger alphabets. First we show that the ternary rich sequence with minimum critical exponent is a morphic image of a fixed point, i.e., an HD0L sequence. Second we draw attention to the fact that the rich sequences having the minimum critical exponent show a large degree of symmetry, i.e., they are G-rich with respect to a group G generated by more than one antimorphism. The notion of G-richness generalizes the notion of richness in palindromes which is based on one antimorphism, namely the reversal mapping.