Are the holes on the undersides or lower side area of the rhizomes? Are they circular? If so, they are where old roots used to be. That would be OK.
I have never heard of a slug eating a living rhizome, so I don't think its that.
And I am sure you don't have borers at this time of year...all should be pupa by now.
Laetitia

Sounds like crickets to me. For some reason if the rhizome is exposed they like to munch on them here, like you said small indentation with sawdust. I have been putting either cayenne pepper or garlic powder (not salt) on or around the rhizome and then maybe cover it with some sand. They usually leave it alone after that and go to another. When I first discovered the holes I searched the fans, dug around the rhizomes--nothing, then I decided to go check at nite and caught them in the act red handed.

The holes are on the top surface of the rhizome and these are brand new plants. I inspected all of them a week ago when they arrived and no holes were present.

Crickets... now THAT sounds like a winner. We've got a bunch of them hopping all over the place. I thought of the sand idea but wasn't sure if, by placing some on the rhizome, I might encourage rot. I guess these critters, by boring these holes, are creating conditions far worse.

I sprayed all the plants with some insecticide before coming in tonight. And, thinking it might be slugs, I put a few saucers out with beer. I just checked and there were scores of crickets in them!

Sand over the rhizomes + some more spraying in/around the bed might be my best bet.

I can't believe I've got to mess around with crickets!