1911 eyedropper mabie Todd leaks

I have recently purchased a 1911 hard rubber eyedropper mabie todd. The pen is in goos condition but it leaks a little ink between the barrel and the section and it stains my fingers.
What would you advise? Sylicon grese? Can sylicon grease be applied to hard rubber?
Maybe a general discussion about vintage eyedroppers leaks is interesting for all