@PierrePC and ThomasKojar, thanks for the pointers. This gives me quite a bit to go on. The Stroock-Varadhan-type support theorems seem pretty complicated in terms of their conditions, so I'm still hopeful there's something simpler that would ensure full support, even if it's not as generally stated as possible (the constant diffusion coefficient might help here also).

I don’t know of one, but I’m not very familiar with the information geometry literature.

Correct (provided one handles the indeterminate case mentioned).