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If he was buffing I would think he would continue betting.
There is a small group of weak draws in a good flop check raise range that should fold to a flop 3 bet, the bulk of which should be checked on turns that benefit the flop bettor(I think the J turn fits this criteria). If the big blind chooses to continue betting on the turn with every draw, he would be quite draw heavy and thus is vulnerable to frequent turn raises and light calldowns.
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Flop seems like a check and fold brick turns unimproved.
I would check this hand sometimes on the flop, in which case I would also fold the turn unimproved.
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As played I would bet this turn card.
Every Queen in our range has an open ender. I think that betting every Queen would be too many draws in the betting range, thus I would check some Queens in addition to all of the dumb end straight draws.