Coincidence of invariant measure for the alternate base transformations

We characterize all pairs $(\beta,n),(\beta^\prime,m)$ such that the alternate $(\beta,n)$ and $(\beta^\prime,m)$-transformations $K_{(\beta,n)}$ and $K_{(\beta^\prime,m)}$ have the same absolutely continuous invariant measure, where $K_{(\beta,n)}(i,x)=(i+1 \mod 2 ,T_i(x))$ with $i\in\{0,1\}$, $T_0(x)=T_\beta (x)=\beta x \mod 1$, $T_1(x)=T_n(x)=nx\mod 1$ with $\beta>1$ real and $n\geq 2$ an integer.