What's the market saying about forward inflation?

Here's a picture of 1-year forward 1-year inflation.

1 year forward 1 year inflation

It's risen but it's not arguing for problematic inflation.

The two-year forward picture has been turning lower, presumably as the market prices in fewer rate cuts.

2 year breakevens
2 year breakevens

I think there's a portion of the market that is always worried about inflation and the post-covid environment certainly expanded their numbers. There's always the possibility of another run-up in energy prices or some kind of shipping shock but I really don't see a problem here. Yes, inflation could run at 3.5% instead of 2% and the Fed may even need to hike again but we're not on the brink of some kind of fresh inflation problem.