Gave it a go.
The Rit Dyemore for synthetic materials worked pretty well. Their green is called Peacock Green, which is a darker green. Already close to OD as it was. I mixed it strong, half the bottle to only a few cups of water, with a little vinegar in there as well.
After two minutes, the color was an OD, but with a bit of a muddy brownish look, from the original brown underneath.
An extra minute made for a darker drab green color. A bit darker than typical OD, but I like it.
There is a very little bit of the brown peeking through on the edges, as they don't take dye as well, but it looks good.
I actually liked the result at the 2min mark, but wanted a bit less brown, two and a half minutes may have been perfect for me, but I am happy with the results.