Shitsticks, I was planning to drive up there in early June - had everything planned and everything! I'm going to hold out hope it'll be fine, since I'm already going to be up there regardless.
At least you'll be there in your own vehicle, so the car rental agencies will have no say. My only vehicle now is a Prius, which can't do the Dempster, so my plan was to fly up to Inuvik from Edmonton and rent. The company I contacted is one I've rented from before in another location, and is not fussy about going off-pavement, but is nervous about the road to Tuk in its present condition and less than confident about how much better it'll be in June.
I'm the Travel Mapping maintainer for the Northwest Territories, and in December added to TM the road to Tuk as NT route 10. Assuming you travel the highway this summer and I end up not going, any comments would be welcome, especially route signage and where it begins (at the Marine Bypass Road junction, or farther north as I have it?) and ends.
Also, pay attention to the two ferry crossings on the Dempster Highway south of Inuvik. Right now, one is closed for the season; the other is not, but subject to short-notice closures. On average, both are open by June 2, but if your luck is bad you won't even be able to get to Inuvik.