Diagonal of (self) product of doubly stochastic transition matrix

By doubly stochastic and transition, I mean each row sum and column sum of a matrix is 1 and each element of the matrix is in [0, 1]. Here, I am considering the matrix is n by n where n is finite.
I’m curious to know that if P and Q are doubly stochastic transition matrices, can we say something about the diagonal elements of PQ? (something like they are positive.)
What about the diagonal elements of $P^2$ where P is a doubly stochastic transition matrix?
(Eventually, I want them to be greater than zero to show that for discrete-time Markov chain with doubly stochastic transition matrix and finite state space, all states are recurrent.)

Edit: As suggested by @kimchilover, there is an example that doesn’t do what I want a doubly stochastic transition matrix to do. However, the particular example gives an irreducible finite MC, so it eventually does what I want to show which is that all states are recurrent. Is this always the case? If it is, how do I go about showing this?
Thanks!