Well... This would take some work. But it could be doable. Number the entries, say, #1 to #28 like we had in this contest. Group reviewers into teams of four.
Team Member A starts at the top and does odd reviews first, so, #1, #3, #5 and so on. Once those are finished xe does the even reviews in the same pattern.
Team Member B starts at the bottom and does odd reviews first, so #27, #25, #23 and so on. Once those are finished xe does the even reviews in the same pattern.
Team Member C starts at the top and does evens first, #2, #4, #6... Then does the odd reviews in the same pattern.
Team Member D starts at the bottom and does evens first, #28, #26, #24... And then does the odd reviews in the same pattern.
Thus, every reviewer only needs to do seven entries for each person to get a review, and that can be done in one or two sittings with minimal stress. And each contestant will ultimately get four reviews.
Just a thought. It's a bit late to implement for this contest, but I'd be down for doing it with the other people who are operating #2.