I can't make an argument for not folding this hand in any step of the way.
This was interesting for me, as I was also surprised by the equity this hand has on a flop without broadways and "unfiltered" villain range. If it had happened 2/3 days ago, I'd likely let it go easy.
I will try to go through the thinking process, street by street, and would appreciate you point the flaws.
PF
Considering BTN stndard OR, we are:
Equity Vencer Empate
BU 51.74% 51.17% 0.57% { 22+, A2s+, K2s+, Q6s+, J7s+, T8s+, 98s, 87s, 76s, A2o+, K7o+, Q9o+, J9o+, T9o }
BB 48.26% 47.68% 0.57% { 3h3s }
Raising here will transform it to something like:
Equity Vencer Empate
BU 54.98% 54.22% 0.75% { JJ-22, AQs-A2s, K7s+, QTs+, JTs, T9s, 98s, 87s, 76s, AQo-ATo, KJo+, QJo, JTo }
BB 45.02% 44.27% 0.75% { 3h3s }
So, basically, in my interpretation, we would be getting 3%FE by narrowing villain range, that would translate in equity loss.
(I don't know if I can assume the FE to be the % of villain range villain would drop to a 3bet).
Adding on the above, unless we hit a 3 OTF, there would be no "easy" way to play:
A flop with broadway cards will "force" us to range CBet into the same range villain call us with.
A "low" flop like this we have low to no FE and on the 50% equity border, we need it.
Flop
Now, keeping villain range, this flop and basically any without broadway cards [1] we have a quite nice equity:
Board: 9♥2♥4♠
Equity Vencer Empate
BU 45.84% 45.71% 0.13% { 22+, A2s+, K2s+, Q6s+, J7s+, T8s+, 98s, 87s, 76s, A2o+, K7o+, Q9o+, J9o+, T9o }
BB 54.16% 54.02% 0.13% { 3h3s }
But, there are no runouts that we like. Meaning we have a fair amount of equity, but we will have an hard time realizing it. This is why I opted for the check raise. I think the check/raise will grant me more FE than any other action here. I accept I can be very wrong here.
Ofc, if we are called, we just have to say goodbye to the pot, as our equity against calling hands would be something like:
Board: 9♥2♥4♠
Equity Vencer Empate
BU 79.43% 79.43% 0.00% { TT+, 9d9s, 9d9c, 9s9c, 4d4h, 4d4c, 4h4c, 2d2s, 2d2c, 2s2c, AhKh, AhQh, KhQh, AhJh, KhJh, QhJh, AhTh, KhTh, QhTh, JhTh, Ad9d, As9s, Ac9c, Kd9d, Ks9s, Kc9c, Qd9d, Qs9s, Qc9c, Jd9d, Js9s, Jc9c, Td9d, Ts9s, Tc9c, Ah8h, Kh8h, Qh8h, Jh8h, Th8h, 9d8d, 9s8s, 9c8c, Ah7h, Kh7h, Qh7h, Jh7h, 8h7h, Ah6h, Kh6h, Qh6h, 7h6h, Ah5h, Kh5h, Ah4h, Kh4h, Ad9s, Ad9c, Ah9d, Ah9s, Ah9c, As9d, As9c, Ac9d, Ac9s, Kd9s, Kd9c, Kh9d, Kh9s, Kh9c, Ks9d, Ks9c, Kc9d, Kc9s, Qd9s, Qd9c, Qh9d, Qh9s, Qh9c, Qs9d, Qs9c, Qc9d, Qc9s, Jd9s, Jd9c, Jh9d, Jh9s, Jh9c, Js9d, Js9c, Jc9d, Jc9s, Td9s, Td9c, Th9d, Th9s, Th9c, Ts9d, Ts9c, Tc9d, Tc9s }
BB 20.57% 20.57% 0.00% { 3h3s }
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With his 3bet call range:
Board: 9♥2♥4♠
Equity Vencer Empate
BU 45.14% 44.91% 0.23% { JJ-22, AQs-A2s, K7s+, QTs+, JTs, T9s, 98s, 87s, 76s, AQo-ATo, KJo+, QJo, JTo }
BB 54.86% 54.63% 0.23% { 3h3s }
If we had 3betted I think we would have a more difficult spot to navigate, specailly because we wouldn't be able to represent anything in a board that "is good for us", but terrible for our perceived range.
We wouldn't have more equity, I doubt we would have more FE (actually think is the opposite) and we would have a bigger pot (thus difficul to play).
[1] Actually surprising equities from low pocket pairs on textures like this. I reccomend you run some experiences in equilab.
It surprised me the 45% to 55% equity situations one lands in, that imo are spots where we should apply some pressure to realize our equity and deny villain's, specially on micros. If facing resistence, fold.
Not sure if it added anything else to the discussion.