Finishing an old no-fretboard neck

Hi folks:

I'm refinishing the neck of an old 1890s no-name spunover-pot type banjo. The finish on the back of the neck has worn off, so it is time... This is a nice playing, non-collectible instrument. I'm just interested in respectfully keeping it in play.

This is one of the no-fretboard type instruments. (It doesn't even have the paper-thin ebonized wood fretboard; the frets go right in the neck.) The "fretboard" wood is darkened however.

Do I need to put any sort of finish on the "fretboard" area? I was planning on using Fiebings (alcohol based) dye to blacken it more evenly. On top of that, do I need a thin finish? Tung oil? Tru oil? Wax? Nothing?

Thanks!

Chris