Ok, so, same resolution, but going for very high frame rates. In that case, you want to have the fastest GPU for your money.
I will change what I say in terms of when to buy.
Get the monitor first. Since it has FreeSync (range 40-144) and is GSync compatible. Further, it has Low Framerate Compensation, so, if needed, it could go all the way down to adjusting to 20fps and maintaining smoothness.
I'd say, with the GSync enabled in the drivers, try out your 1060 with this monitor. The FreeSync capability will allow things to smooth out as the framerate dips for any particular game. You might find it an enjoyable experience with your existing card.
If you're still not satisfied at that point, and what much higher frame rates to take advantage of the high refresh rate this monitor, then I would suggest getting a 5700 XT, since it's 2070 performance at 2060 Super's price.