Question Skybox isn't changing in reflections?

I've got a skybox with an animated shader, a reflection probe, and a shiny object. The reflection of the sky on the shiny object updates in the editor, but it doesn't update during the game.

Things I've tried that don't work:

• Setting the reflection probe to Refresh Mode: Every frame has no effect.
• I tried setting it to Refresh Mode: Via scripting and calling its

  RenderProbe()

  method, but that doesn't have any effect either.

The one thing that does work:

• Changing the skybox material's inputs in the editor -- this causes the reflection of the skybox to update, but only while editing, not while in game mode.

Any ideas what I should try next?