Nothing that Indigo uses directly is stored on the external drives. I do use the Plex plugin and store my video files on the HDs, but I tested with the plugins disabled and the issue was still there. Time Machine backups are networked to my Mac mini running Server.
So I tried another test today. I spun up the external hard drives before connecting via the client, and the issue did not occur. I've tested this twice now.
I'm using a MacBook Air, and even though I've set Energy Saver not to spin the drives down I'm certain they are going to sleep.