I think the basins will continue being topped up by warmer waters from the south as the cooler lighter layers go south, through CAA. The Gulf stream imho is a residual of the tidal cycle which will continue to cycle whatever water is present, whether thats from the Gulf or, seasonally, Arctic waters from Baffin/Greenland.

It's a pretty graphic, but I don't think it makes physical sense to say that tidal forces can permanently create sea water circulation, with surface waters moving towards the poles and deep waters moving towards the equator. It takes massive amounts of energy to force the world's oceans to circulate in observed patterns, and the tidal forces from gravitatioal effects don't seem to be adequate.
The conventional model is that ocean circulation is a heat engine effect. Solar energy warms surface waters at the equator, and polar water cools and sinks, creating a kind of down-hill gradient for surface waters to move poleward.
Much of the tendency for polar waters to sink derives from ice formation, with brine exclusion causing cold, salty water to sink. It makes more sense to see the Gulf Stream poleward directional flow as being driven strongly by the total volume of annual ice formation in the arctic.
When the arctic is perennially ice-free, there will be much less heat engine efficiency in moving Atlantic circulation. This is a negative feedback mechanism that will tend to retard the final disappearance of winter ice.
Edit:
I found the above graphic at:
https://www.uwgb.edu/dutchs/EarthSC102Notes/102TheOceans.HTMIt indicates that the graphic relates to vertical (tidal) motion, not bulk horizontal motion. A separate section describes ocean circulation.