prove/disprove if each two in $n$ operators can be diagonalizable simultaneously then all can be diagonalizable simultaneously

Question

I have an idea that for $n$ diagonalizable operators $A_1, A_2, ..., A_n \in \ell(V)$. if each $A_i, A_j$ can be diagonalizable simultaneously then all of them can be diagonalizable simultaneously.

if it is true then we can prove that for every permutation the answer of $A_{i_1}A_{i_2}...A_{i_n}$ is the same iff they are diagonalizable simultaneously.

can you prove or disprove that?