Monday & Tuesday: after cleaning out the closet, I spackled, patched, and caulked. There were many holes in the walls from old hanging systems that were no longer extant (as well as an old, capped gas line you can see near the middle of the picture). I also sealed the gaps in the drywall box (the left edge of the picture) that houses more HVAC ducts. This ought to help cut down on the dust in the closet.
I created some additional shelf supports with scrap wood left over from my basement demolition and painted the whole space with primer.