The problem is not correctness, but whether the signal is properly received by your audience. You chose 'his or her', but was this a conscious choice? And at what level? Was it because you fear being judged poorly for the grammatically correct but non-politically correct 'his'? Because you find 'eir' pretentious? Because you find substituting 'her' to be an affectation?
To me, 'His or her' is a signals that you are aware that language can be sexist, and want to show that you are not. I think it correlates with college-educated American liberal born after 1960, or one who has learned his English from such.
I fit these characteristics, but usually consciously choose 'his', hopefully signalling that I detest linguistic contortions for the sake of signalling political correctness, but more likely just being judged an unrepentant sexist. If I don't want to take this risk, I switch to an across-the-board 'her', as I feel this is more effective at actually combatting sexism.
Have you read Hofstadter's essay on "Purity in Language"? http://www.cs.virginia.edu/~evans/cs655/readings/purity.html