If this soliton behaves like a material with a negative mass, does it also have a negative gravity?

Amazing and wonderful. Dr W Brown In The Beginning has a new quandary.

It's way to small to have gravity that can be measured.

If this soliton behaves like a material with a negative mass, does it also have a negative gravity?

yes – but solitons are virtual particles on a background, a soliton with negative mass only means reducing mass density locally, and its negative gravity field only means reduced compared to gravity field of background sea.