As far as I know, courts in the past have ruled that being gay is not an "immutable or highly visible trait." There were however some recent decisions that questioned that rationale.
It reminds me of how "don't ask don't tell" seems fairly reasonable until I imagine a bunch of soldiers away from home sitting around together, waiting for things (which they do an awful lot). Then I try to imagine navigating hundreds or thousands of hours of those conversations while somehow avoiding revealing my sexuality.