Sorry for the late reply.
You need to punch the images at the start to load a level.
Sorry, for the late answer as I was busy recently.
So, the level selection should looks like this : https://gfycat.com/firstgiantivorybackedwoodswallow
Are you playing through the Oculus App or SteamVR ?
The game was developed using SteamVR, so it might not work properly if you are using the Oculus API.