I don't think there's any situation in which you wouldn't start a sentence with a capital letter. Slang, dialect, etc. included.
Hm, how about:
iPods are convenient.
Would that have to be:
IPods are conventient? (Yuk!)
I'm not saying that you can't start the sentence with a capital letter in this case. I'm just saying it wouldn't be my preference. (Similarly, I'd probably write: "e.e. cummings is my favourite poet," rather than "E.e. cummings is my favourite poet.")
I accept that other people may have other preferences, but to me propernames that start with lower case letters override the sentence initial rule; I like to preserve the original "rule violation" and pile another on top, rather than mess it up with "correctness".
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But in the apostrophe case, I concur. Though I would accept the argument that the absent letter is capitalised: a capitalised apostrophe, so to speak. In other words, I wouldn't mind lower case too much, even though it's not my preference.