We have the arrowhead plant all over the place in Central Ohio along with the narrow leafed arrowhead. I've always found the Latifolia to be semi-aquatic/emergent and never fully submerged. So I guess like submerged trees, they might continue to live until they die from the conditions that they're not used to or fully adapted for. Time will tell I guess.
I tried and failed to get the tuber on this plant. It has a good sized, long hank of roots on it, but the tuber broke off, so I surmise that it's taking in the nutrients of the substrate from food and fish waste. Hopefully the plant will live, if not, well, it was a good experiment to see if emergent/semi-aquatic plants could adapt to a submerged life.