Having the kings indian as my main opening in response for 1.d4 I see nothing immediately loosing (I might be wrong) and would personally be happy to play this position. The variations I have considered are hopefully inline with my argument that black is doing fine and If I have missed something obvious or have misrepresented white please let me know.
Variation 1.
r4rk1/1p1q1pb1/1n1p1npp/pPpPp3/P1P1PPb1/2NQ2P1/3N2BP/R1B2RK1 b Qq - 0 1
1... exf4 2. gxf4 Nh5 3. h3 Bd4+ 4. Kh1 Qe7 5.hxg4 Qh4+
From here you will at least get a rook for two minor pieces after white blocks and knight g3+. More importantly we can note that white has not done anything in the last 4 moves that could be considered active, so I would think the tempo gained for the bishop is worth it.
Well, white took black's bishop with the h-pawn... perhaps another move is possible, say going after the monster bishop on d4. (note in the following that giving up the rook on a1 i.e moving the knight on c3 to attack d4 is horrible for white);
Variation 2.
r4rk1/1p1q1pb1/1n1p1npp/pPpPp3/P1P1PPb1/2NQ2P1/3N2BP/R1B2RK1 b Qq - 0 1
1... exf4 2. gxf4 Nh5 3. h3 Bd4+ 4. Kh1 Qe7 5.Nb3 Qh4 6. Nxd4 cxd4
Here black is looking to be an exchange up with Ng3+ which is made possible by the knight on c3 being under attack. I would say white was better in variation 1 then here.
A major flaw in the above variations have been kh1 as it has set up the main Ng3+ ending for black which have reduced material inequality. So what about kh2:
Variation 3.
r4rk1/1p1q1pb1/1n1p1npp/pPpPp3/P1P1PPb1/2NQ2P1/3N2BP/R1B2RK1 b Qq - 0 1
1... exf4 2. gxf4 Nh5 3. h3 Bd4+ 4. Kh2 Nxf4 5. Rxf4 Be5 6. hxg4 Bxf4+ 7. Kg1 Qxg4
Here black is up a rook and two pawns and some tempo for two minor pieces. A drawback is of course having only two active pieces, so f5 must probably be played soon if black is to win/draw.
Now the question arises if 2. gxf4 is any good for white... perhaps instead 2. Rxf4:
Variation 4.
r4rk1/1p1q1pb1/1n1p1npp/pPpPp3/P1P1PPb1/2NQ2P1/3N2BP/R1B2RK1 b Qq - 0 1
1... exf4 2. Rxf4 Nh5 3. Rf1 Bd4+ 4. Kh1 f5 5.exf5? Bxf5
- exf5 seems way to dubious for white as it just improves the bishop for black and the two open files are easier occupied by black's rooks.
Variation 5.
r4rk1/1p1q1pb1/1n1p1npp/pPpPp3/P1P1PPb1/2NQ2P1/3N2BP/R1B2RK1 b Qq - 0 1
1... exf4 2. Rxf4 Nh5 3. Rf1 Bd4+ 4. Kh1 f5 5.h3 f4 6. hxg4 Nxg3+
As @John Coleman points out in the comments I have been partial to h3. Therefore I am adding some more variations in which Nf3 is played to which I believe the key response is Bxf3;
Variation 6.
r4rk1/1p1q1pb1/1n1p1npp/pPpPp3/P1P1PPb1/2NQ2P1/3N2BP/R1B2RK1 b Qq - 0 1
1... exf4 2. gxf4 Nh5 3. Nf3 Bxf3 4. Bxf3 Qh3 5. Be3 f5
Variation 7.
r4rk1/1p1q1pb1/1n1p1npp/pPpPp3/P1P1PPb1/2NQ2P1/3N2BP/R1B2RK1 b Qq - 0 1
1... exf4 2. gxf4 Nh5 3. Nf3 Bxf3 4. Qxf3 Nxc4
Variation 8.
r4rk1/1p1q1pb1/1n1p1npp/pPpPp3/P1P1PPb1/2NQ2P1/3N2BP/R1B2RK1 b Qq - 0 1
1... exf4 2. gxf4 Nh5 3. Nf3 Bxf3 4. Rxf3 f5
Each of these variations need analysis in themselves. Variation 7 does not look appetizing at all for white and can probably be disregarded. In variation 6 it looks like black has trapped their own queen after say 1... Bg2 however 1.. Qxe3 2. Qxe3 Bd4! pins the queen to the king and all is good. Variation 6 and 8 looks decent for black with the clear goal of bringing the rook from a8 to e8. While white also has a decent position in those variations, there are some concerns with defending c4 with some piece other than the queen and getting the a1 rook into the game.