The way I understand tinnitus, I don't think it would make a difference.
Here's what I know: Our inner ears contain hair cells which would normally be responsible for perceiving specific frequencies when stimulated by the basilar membrane inside the cochlea (which is simulated by the 3, tiny bones which are stimulated by the attached ear drum, which is stimulated by...you get the idea.). With tinnitus, however, some of these hair cells are damaged and can no longer detect vibrations. As a result, the accompanying neurons associated with those damaged cells become "hungry" for stimulation because the brain sends an increasing level of "outbound" signal since it never receives any "inbound" signal, thereby causing the ringing sound we hear...a "loop" of information, if you will. This is very similar to the phantom-limb pain we can feel after having lost an arm or leg; this situation also causes the associated part of the brain to stop receiving signals from the amputated or damaged limb and the increased level of outbound signal causes (severe, in many cases) pain. Tinnitus works the same way, but on a much smaller scale because fewer neurons are left wanting, plus we still have many hair cells remaining which function normally and help "drown out" the ringing.
So, by that rationale I imagine that the ringing would be much more apparent initially but would eventually be drowned out by the sounds perceived by the working cells, like heart beat, breathing and digestion. Then when those sounds are not enough, our brain starts creating "phantom" stimulus which causes the hallucinations.
In short, I would think the answer is no.