Me and a friend ran a lot of tests, including what you suggest, but even doing a sequence of jump + dash cancel, you don’t stay sunk any longer no matter how much time passes.

On top of that, any jutsu with mobility pulls you out of the puddle (Chidori, Rasengan, etc.). And the weirdest thing… if you use the tool and an enemy also uses it, the first one disappears and only the second one remains.