I think I'm as confused now as you were back then.
I'm guessing you're on the button - since V is in the CO.
You say he's aggressive, yet he did not raise pf... you did.
He did not re-raise you pre-flop (especially after he sees that it's going to be HU) so we can probably eliminate premium hands from his range.
In our 1/2 games - a half pot bet is often as effective as a full pot bet HU. But on a flushing board, 3/4 to full pot is always best.
So I don't know if you've induced a raise or V genuinely has a monster hand.
For most (especially in our 2/5 games), a check is a sign of weakness.
But in our 1/2 games, a check can be the ultimate sign of strength - and it's a trap.
If he had 2 pair (like J8) or a set, he's now got a boat - so he's can allow the flush to get there and still win the hand.
I think the smooth-call on the flop was good. But now, I'd check the turn.
Think about it... what are you protecting your Kings from?
A Flush? A str8?
Would V check-raise oop with such a draw? IP it's done a lot.
OOP, this seems suicidal.
But it depends on YOUR reads on this V. What hands has he shown down previously? How has he played his big hands before? How has he played his draws? If he's so aggressive, why didn't he bet the flop?
What hand does he put you on?
I don't know why you're so content shoving at any point in this hand.
Even if V has a weak J
x
, you're now a huge dog.
I'd check it here. And unless you improve, even c/f the river.
Unless you've mis-represented your hand (and i know even less about your image than your V), I've got to believe you're beat.