EEVEE SSR interfering with reflection planes

While reflection planes give correct results in this scene when SSR is turned off, as soon as I turn SSR on, it gives weird results, where screen space reflections and nearby cubemaps appear to override parts of the reflection plane.
When I clear the lighting cache, it turns offscreen reflections black, despite the presence of the plane.
Is this a bug?
Is there a way to deal with this without affecting SSR on other objects?

Without SSR:

With SSR (cubemaps active):