Another question on commutativity of special type of permutations

Question

Referring to this Commutativity of special type of permutations , if $p_1p_2=p_2p_1 , p_1 \ne p_2 , p_1 \ne p_2^{-1} $ , then is it true that $U_1 \cap U_2=\phi$ ?

Answer 1

Take $S_6$, let $p_1=(1,2)(5,6)$ and $p_2=(3,4)(5,6)\neq p_1$. One can verify that $p_1p_2=p_2p_1=(1,2)(3,4)\neq e$. But, $U_1\cap U_2=\{5,6\} \neq \emptyset$.