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I agree pushing isn't terrible here. My move would probally be to fold and call a push in the bb next hand.

If I push here I will be called by a hand that beats mine. If I call someone from my bb, I could easily find myself calling an any 2 push from a desperate shortstack or loose large stack, which will be ok considering my odds.

Maybe it's also better for image. You are kind of expected to call from your bb, but pushing with J7 UTG shows you are capable of pushing trash.

I'm not sure you have much FE at all. Shortie in the bb, another shortie who will have an itchy trigger finger, and a couple of large stacks who might challenge a UTG push from a shortstack.

There's also always the chance that your BB is folded to you.

No matter what you do it's a $hitty situation.

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J7 is an average hand. Next hand you may have a worse one. You will be forced to call bets from hands that will on average be stronger than the hand you have. Saying the short stack will be pushing worse hands than normal is irrelevent as you have no choice other than to call regardless of your hand. They have a choice so will weed out the very worst hands that they will move with. Their range will always be better than yours.
By pushing now you have some FE, not much, but 100% more than you have leaving it until the next hand.
Push