Game now fully working in WebGL

It turns out the reason why the game wouldn't load in browser was that Unity decided to enable compression by default.
So simply disabling compression in Unity's settings fixed the issue.

Go try out the game, now that it fully works in browser!

And those who had a similar issue, here's a video that shows how to fix it:
https://www.youtube.com/watch?v=2jjESP58jsA