It appears likely that we've already seen the last of the National League without the designated hitter. As part of the owners' proposal to play the 2020 season (a proposal that appears to have a good chance to turn into a deal), the universal DH would be implemented for both the 2020 and 2021 seasons, per Joel Sherman of the New York Post.
After the 2021 season, the collective bargaining agreement between the owners and players expires and then they'll need to put together a new one. After two years with the DH being used by both teams in every single game, I see no realistic reason that the universal DH wouldn't be included moving forward on a permanent basis.
Why? The explanation is rather simple.
The owners have already proposed this, meaning we know they are on board. The players wouldn't turn down the extra opportunity to hang onto a job once one's defensive chops have eroded -- and it's also likely they can get rosters to expand to 26 or 27 men on a permanent basis as part of these talks -- especially since many DHs are veterans who have loud voices in the MLBPA.
Also, having both leagues play by a different set of rules only to have to adapt in the World Series really doesn't make sense. Obviously, the players are far more in favor of adding jobs than subtracting.
Basically, this seems like a done deal. Fans who hate the idea of the DH better starting getting used to the idea. This isn't going away. We'll have the universal DH the next two seasons and then the most likely outcome is it being included in the next CBA.