Is there any reason to interpret the coefficient by itself, rather than relative to other coefficients of other areas/times? If you were comparing coefficients you would of course use the same real-adjusted currency. If you aren’t comparing, I don’t see the point of the coefficient nor the problem with it changing due to the income units used.
I’d have thought you’d want to be able to say things like “if the coefficient is at least 0.618033989 then you should give very serious thought to whether you have a problem”.
And it’s all very well to say that if you were comparing then of course you would use a common currency for both—but in practice I doubt it would work that way. If you compare two countries’ GDPs, you look them both up, do a single currency conversion, and compare the numbers. You can’t do that with logini; if you know that a country’s logini is 0.739085133 when computed using pounds sterling in 2017, that is not enough information to convert it into US dollars in 2020 (even in 2020 when you know how much the dollar is worth then).
I suppose that if logini coefficients took off then people computing them might pick some standard-for-all-time currency and convert to that. Which is more or less equivalent to my suggestion of using something like the price of a loaf of bread as a unit, except a bit more arbitrary.