I highly doubt it. Nintendo unjustifiably failed to support HD, and they actually would be justified in not supporting 4K. Granted, you could make an argument that there's some overlap in the 4K and hardcore gaming markets ("techies" with high disposable income) but that's still not a large market. Then you have people with 4K sets that are starved by 4K content, but how big that market? Are they interested in a dedicated gaming machine? How does Nintendo win those people over?
Too much cost, too many questionables for NX to support 4K.
I think you are overestimating the cost of 4K. The requirement to do 4k are the base specs that a NX console should support. This is not solely based on making games produce a 4K res, but its the natural progression of the technology, at a bare minimum for a console launching in 2016. Nintendo would have to actively downgrade the system to make it less powerful. Mind you, this is all based on the system being "mostly" tradition in terms of its market aim.