1) Dont use anything above 5W30. And use ILSAC GF-6/6A Oil.
(you have already shared results of using 10w40). thicker oils are harmful for these engines.
2) Try to loosen gum between piston rings by using engine oil cleaning solution (liqui moly etc.) but in ur case u have mentioned that rings have been replaced.
3) Does oil consumption increases with driving in high rpm constantly (above 2.5k rpm and high speeds)?
4) in i-DSi i dont think there is a conventional spring loaded vavle. If there is then try cleaning. (also share pictures)
I am having same problem with Honda City vario 2006, 200k driven. The oil gets consumed when driven on highway at high speed.