A necessary condition is that, for every word $w(x,y)$ in two letters, we have $${\rm Tr}w(A_i',A_i)={\rm Tr}(B_i',B_i).$$ If $r=1$ (one pair only), this condition is also sufficient; this is Specht's Theorem. Notice that the symmetry assumption is not important here.
Denis Serre
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