With any luck he will.
A suggestion for getting a second filter established quickly, if I may ?
Clean the first filter in the smallest amount of used tank water possible. Squeeze out the media well, get as much of the dirt and crud into the water as you can.
Then pour that dirty water into the new filter on the new tank. It should be cycled within a day or two, and so long as you have something alive in the tank or feed a bit of ammonia, the BB will continue to grow on. This is faster than putting a second filter on the first tank and letting it filter.. because not many BB float free in the water column. A few get knocked off their surfaces, but by seeding it with the 'squeezings' from an established filter, you dramatically increase the number of BB that go into the new filter. It's a serious jump start. I've done this and been able to put fish in the second day.. and it's how I set up isolation tanks in a hurry if there's an emergency.
Some keep an extra sponge in a filter so it's available to set up a tank in a hurry, for any reason that may come up. It's an easy way to get a cycle in almost no time flat.