Given regular matrices $A_i,B_i \in \textrm{GL}_n(\mathbb{R}),$ $i=1,2$.

Let $A_1 = U_1 B_1 V_1$ and $A_2=U_2 B_2 V_2$ where $U_i,V_i \in \textrm{O}_n(\mathbb{R})$ $(i=1,2)$ are orthogonal matrices. This means that $A_1$ and $B_1$ resp. $A_2$ and $B_2$ have the same singular values.

Now let $A_2 A_1^{-1}$ and $B_2 B_1^{-1}$ have the same singular values.

I assume that in this case one can find $U_i',V' \in \textrm{O}_n(\mathbb{R})$ so that $A_1 = U_1' B_1 V$ and $A_2=U_2' B_2 V$.

Is this true, at least under some conditions? Or is there a simple counterexample?