Recently there was a nice question asked on convexity of American put w.r.t strike: Convexity of an American put option
Does the same hold for a Bermudan option in rates, where they underlyings are different swap rates? Say you have a Bermudan callable every one year from now for the next 5 years, and the underlyings are fixed-float swaps which have a tenor+maturity=10y (i.e, if I exercise at 1y, I enter into a 9 y swap, if I exercise at 2 y, I enter into an 8y swap..).