Darken specular highlights from a certain light source

I’ve got the following problem:
I have a shiny room floor with reflections and a sun light (directional light). I want the interior darkened but still lighted by the sun (this is an aesthetic choice, not one out of realism… but GI doesn’t get enough light inside if I’d rely on that). I put a semi-transparent quad with shadows only to darken the specific areas. Now I have the problem that I get very bright specular highlights from the sun on my shiny floor. Is there any way to keep the reflections but get right or dim down the specular highlight?

Or do you have any other ideas to achieve the target result differently?

Sorry, I can see very clearly that this post is from 4 years ago, but I’m having a similar issue now. On reflective surfaces, the “sun disk” specular from directional lights is disproportionately bright. This makes it such that the sun in the sky is affected slightly by bloom, while the reflection is much more intense.

Have you, by chance, found any sort of solution to this in the 4 years the question went unanswered?