One thing that seems to be pretty clear id that the whole camera market is in a move at the moment.
If the DX plays a role after the market has stabalized depends if the customers (us) are still buying it. If there is a demand then there will be companies serving the need.
The problem I see is the purpose use of the DX system. As I see it there are three main camera purposes depending what you are going to shoot. Purposes in this term are planned, semi-planned and unplanned.
For planned shooting - so you are going to shoot a very specific moment or motive and even travel for the same, if you are pro or amateur, you want to have the best possible picture. For this type of shots, with the FX prices dropping, FullFrame cams, if the budget allows, will be your tool - now and in the future. With or without mirror or what ever other gimmick the vendors come up with is not that important in this term.
For the unplanned shooting in street, capturing the unexpected, users usually want something fitting into the pocket. Traditionally this had been covered by pocket P&S cams. This market is now moving to the smarthpone buisiness. So the P&S cams, even so thrown at high evolution rate into the market (and eating up vast amounts of the vendors development costs) will vanish i think. The same in this term applies to the video digicams sharing the same future I think.
The interesting market so is the semi-planned, travel and vacation market. Cams should be good enough to deliver decent image quality without burden too much when carrying. Being a APS-C terotory in the past m43 and other mirrorless techniques are gaining shares here now. The question is if they can keep up with the demands. The m43 system is stuck on a 16Mpix resultion for now and is there since quite some time. While this is good enough for most use cases the general trend is heading for more. I do not think M43 will be able to go higher then this without revolutionary new technology (Foveon or what ever).
This I see as a chance for the DX system. If the vendors could slim down the size of the cams without changing the flange distance - so the existing lenses COULD be used and then start to slim down the lenses a little (making them DX only) while improving their quatity at the same time (Most DX optics are not just smaller but also lower in quality as their FX brothers) then this system will have a future. If not then FX on the high end and mirrorless on the low will nibble bites off the market until it is vanished.