I wasn't even sure that pushing on a safe turn and hoping for a split was justified as far as odds go. Assuming we add someone with the same str8 as you that steals a flush and boat out, you are playing for 1/5 of a 4-way pot on the flop:
http://twodimes.net/h/?z=1617805
pokenum -o 9h th jc 8h - ad 6d 5h 4s - kc kh 7s 3d - tc js ks 8d -- qd kd 9c
Omaha Hi: 528 enumerated boards containing 9c Kd Qd
cards win %win lose %lose tie %tie EV
Jc Th 9h 8h 1 0.19 312 59.09 215 40.72 0.205
4s Ad 6d 5h 136 25.76 392 74.24 0 0.00 0.258
7s Kc 3d Kh 176 33.33 352 66.67 0 0.00 0.333
Ks Js Tc 8d 0 0.00 313 59.28 215 40.72 0.204
On the turn, assuming a safe card, you are in much better shape:
http://twodimes.net/h/?z=1617819
pokenum -o 9h th jc 8h - ad 6d 5h 4s - kc kh 7s 3d - tc js ks 8d -- qd kd 9c 3s
Omaha Hi: 32 enumerated boards containing 3s 9c Kd Qd
cards win %win lose %lose tie %tie EV
Jc Th 9h 8h 0 0.00 13 40.62 19 59.38 0.297
4s Ad 6d 5h 6 18.75 26 81.25 0 0.00 0.188
7s Kc 3d Kh 7 21.88 25 78.12 0 0.00 0.219
Ks Js Tc 8d 0 0.00 13 40.62 19 59.38 0.297
Now you're ahead of the game and a push is odds-worthy. Until I see something else, I think we assume that Bri's line may be the best from a math standpoint. If you include some potential fold equity it would jump up a bit more I'm sure. Of course, if you get heads up with the one caller you don't want to see (same str8, with flush draw) then you are not happy:
http://twodimes.net/h/?z=1617828
pokenum -o 9h th jc 8h - tc jd ks 8d -- qd kd 9c 3s
Omaha Hi: 40 enumerated boards containing 3s 9c Kd Qd
cards win %win lose %lose tie %tie EV
Jc Th 9h 8h 0 0.00 9 22.50 31 77.50 0.388
Ks Tc Jd 8d 9 22.50 0 0.00 31 77.50 0.613
Now you're taking it in the can more often than you'd like.
To clarify my line, I would generally min raise there if I raised. That is probably not ideal, but at least it would give me a better idea where I stand. If I end up folding the nuts, that's something I can live with.
FWIW, here's Monk's assessment of str8s on his BTP FAQ page:
REMEMBER that straights are VERY VERY WEAK. On a GOOD table, where re-raises are almost always the nuts, you may now and again actually FOLD A NUT STRAIGHT ON THE FLOP! Here is an example: Flop brings 8d Ts Jd and you have 9s Qh 7c 4d on the BB after a few limps; you have the nut straight but should FOLD to a solid player re-raising the flop. He almost certainly has the same straight if he’s solid and may have a redraw to a bigger straight flush, or boat. If he does, you are potentially putting your WHOLE stack (say, $100) in the middle to SPLIT a small pot (maybe only $3-10) that’s currently in the middle, when your opponent could have a FREEROLL to win the whole thing! You could be risking $100 to win $3; not smart. Again, at the lowest levels, it’s not easy to put someone 100% on the nuts here so it is more correct to play it out.
I guess you'd have to figure out what he means by 'at the lowest levels' before you decide to proceed. I know for sure that I run away at a $100 table to that action.