The question mark is incorrect because the sentence is an indirect question and not a direct question. Neither of the other answers provides any authoritative source that recommends using or not using a "?" for sentences such as these, but here's what one trustworthy web page says (and I agree):
"Be careful not to put a question mark at the end of an indirect question. [My emphasis.]
The instructor asked the students what they were doing.
I asked my sister if she had a date.
I wonder if Cheney will run for vice president again.
I wonder whether Cheney will run again.
Be careful to distinguish between an indirect question (above), and a question that is embedded within a statement which we do want to end with a question mark.
We can get to Boston quicker, can't we, if we take the interstate?
His question was, can we end this statement with a question mark?
She ended her remarks with a resounding why not?
I wonder: will Cheney run for office again?