To get this off the Unanswered list:
The question was subsequently posted to MathOverflow, where it was answered in the affirmative. Some useful references from Mark Wildon’s answer:
I'm fairly sure M. Gerstenhaber was the first to prove that the irreducibility result. His paper is: On dominance and varieties of commuting matrices, Annals Math. 73 (1961), 324-348. However the result asked for in the question was already known from Theorem 5 in T. S. Motzkin, Olga Taussky, Pairs of matrices with property L II, Trans. Amer. Math. Soc. 80 (1955), 387-401. There is a short account of this work after Remark 3.4 in the paper by Meara and Vinsonhaler mentioned in SJ's answer.
The last reference appears to be to K. C. O'Meara and C. I. Vinsonhaler, On approximately simultaneously diagonalizable matrices, Linear Algebra Appl., 412 (2006), 39 - 74.