Note: This problem remains unsolved, as of 19 April 2020, so do try it out. 400 rep bounty guaranteed for a correct answer

This a variation of this question by @Gamow

Suppose there are $100$ lions and $100$ zebras. The lions function together as a team, and so do the zebras. The lions place themselves first on an infinite plane, and the zebras place themselves next. Game proceeds turn-wise, one turn for the lion team, one for the zebra team. On each turn, a single zebra or lion (depending on which team's turn it is) moves by up to $100$ metres. Lions win if a single zebra is eaten.

Zebras will try to ensure this doesn't happen. Do the lions' have a strategy that works irrespective of the zebras' one?

P.S. @Veedrac's result is definitely helpful (whether or not you understand the math) if you are attempting to solve this.