Fixed point of maps between product of even dim projective spaces

Question

Is this true ? :

There exist a continuous map from $\mathbb{R}P^{2n} \times \mathbb{R}P^{2m}$ to itself with no fixed point. Here n, m ≥ 1.

I know (by an exercise in Hatcher) that maps from $\mathbb{R}P^{2n}$ to itself have fixed point.

What can I say about the current question?