Separate camera for skybox. Skybox won't show.

So I have one camera to render just the skybox, and another which renders everything but the skybox. They both work separately, but for some reason when they are on together, the skybox doesn't draw. Here are my settings:

Main Camera: Clear Flags - Depth Only, Culling Mask - Everything, Projection - Perspective, Depth - 2, Rendering Path - Forward

Skybox Camera: Clear Flags - Skybox, Culling Mask - Nothing, Projection - Perspective, Depth - 1, Rendering Path - Forward

Any ideas why this wouldn't draw the skybox first, and then everything else on top of it? Right now it acts as though the skybox camera isn't even on (although it definitely is, and changing the Depth will make it show only one camera or the other, so I know they are working).