Would guess the earliest this could happen is 2022 with the new collective bargaining agreement. Players Association loves it because it gives a couple more year for aging fatties to make some coin.
Personally, I hate the DH. It’s designed for unthinking AL fans who like to say, “Ooooh, dinger!”. NL fans tend to be more erudite and love the complexities of double switches to keep Bartolo Colon from swinging a bat with bases loaded and two outs.