RE: Ring height for 44mm objective lens?
I think I went with the high rings on my Howa Varminter with my Nikon Monarch 5.5-16.5x44. I had it mounted locally, and I think the guy at the shop told me that the medium rings just didn't quite get the scope up high enough above the bull barrel. If I were you I'd take it to a shop and give both medium and high rings a try.