Chris, thank you for posting an interesting and important question.
I'm pretty sure that the answer, all other things assumed equal, is that the volume gap will stay the same. Essentially that the volume loss is essentially unchanged regardless of the actual extent of the ice cover.
Reasoning goes as follows, going to first (physical) principles :
Imagine a slab of ice in an ocean centered around the NP.
Now, know that the amount of energy available to melt a square meter of that ice is limited.
Essentially there is just X number of Joule's available to melt ice.
We know that to be the case, since a limited amount of energy enters that location, and a limited amount leaves, and neither of these two values depends on the thickness of the ice.
So that square meter of ice melts a given amount of ice, independent of its thickness.
It is also not dependent on the temperature, since melting ice is pinned down at about 0 C.
Also, the amount of heat available for melt is in first principle not dependent on the location (latitude) of the ice, since it takes a lot more energy to melt (1.5 meter FYI) ice than it takes to warm up the entire column of air above it.
For ice at the ocean-ice boundary things are not too different. There is a limited amount of energy available that the ice boundary extracts from the ocean water, and this is (again, in first principles) independent on the latitude of that ocean-ice boundary.
So the amount (volume) of ice melting out is not dependent on the latitude of the ice boundary nor the latitude of any specific piece of ice.
Volume melt (in first principle) thus depends ONLY on the amount of heat that gets inserted into the Arctic. And that depends on 1) the temperature of the rest of the planet (global warming) and 2) the heat absorbed by land-based snow cover (albedo feedback).
Specifically, albedo feedback of ice loss itself does NOT affect volume melt over a season. Only albedo feedback of (land based) snow cover does.
I hope that argument makes sense, but if not, just check out the PIOMAS volume graph :
and see that these are sinus waves with almost the same amplitude (no albedo feedback within a season).
This also means that there appears to be no physical reason why volume loss would slow down as it gets closer to zero, and thus Prof. Wadham's remark is still relevant :
In the end, it will just melt away quite suddenly