Yup, no filter at all.
The lenses are very durable.
The coatings are very durable unless you take steel wool to them.
The problem with adding a filter is that not all filters are created equal, and are not all optically flat (find one that has a designation of at least 1/4th wave or better advertised) and simply introduce spherical aberration potential, unwanted reflections, potential decrease in contrast from scattering light if its really poor quality, etc. And a BIG filter for a big telephoto lens for birds, well, it should cost a ton if it's quality glass. If it doesn't cost much, well, lots of compromise.
After a session, gentle distilled water and soap solution on a cotton ball, dab, do not scrub, and dry with distilled water cotton ball dabbing. Or a typical lens cleaning solution and dab with cotton.
Very best,