PPT Slide
By its very definition his subgame-perfect strategy gives player I the maximum payoff for any strategy of player II.
A strategy that outdoes the sgp strategy in even one strategic pair would contradict the definition and will not be found. The same must hold for the columns of player I and for any dimension in games with more than two players.
Conclusion: The order of removing dominated strategies is completely irrelevant because it has no effect on the sgp strategy being the maximum.