Is every $ \mathbb{R}P^{2n} $ bundle over the circle trivial?

Are there exactly two $ \mathbb{R}P^{2n+1} $ bundles over the circle?

This is a cross-post of (part of) my MSE question

https://math.stackexchange.com/questions/4349052/diffeomorphisms-of-spheres-and-real-projective-spaces

which has been up for a couple weeks and got 8 upvotes and some nice comments but no answers.

My intuition for thinking both answer are yes is that there are exactly 2 sphere bundles over the circle. The trivial one and then the non-trivial (and non-orientable) one which can be realized as the mapping torus of an orientation reversing map of the sphere. So importing that intuition to projective spaces then the orientable $ \mathbb{R}P^{2n+1} $ should have a nontrivial (and non orientable) bundle over the circle while the non orientable $ \mathbb{R}P^{2n} $ should have only the trivial bundle. For $ n=1 $ this checks out since that projective space is orientable and thus we have exactly two bundles over the circle (the trivial one=the 2 torus and the nontrivial one=the Klein bottle).