"Warm air will rise and will cause the low air pressure inside the bottle." This is only a small effect if the bottle is still uncapped. The pressure of the air in the bottle is still almost atmospheric, as the bottle is open to the atmosphere. The gas is hotter in the bottle, and this, by itself would increase the pressure, but the density, $\rho$, of the air in the bottle is reduced and these two factors work in opposite directions, allowing the pressure to be almost unchanged. $$p=\frac{\rho RT}{M_{molar}}\ \ \ (T \text{= Kelvin temperature},)$$I say that the pressure in the bottle is almost unchanged, because at two points at the same level, one (B) inside the bottle and one (O) outside, the pressure outside will be greater by $h(\rho_{cold\ air}-\rho_{hot\ air}) g$ in which h is the height of the hot air column above B – a difficult thing to estimate. But, as I've said, the pressure difference will be small, in the order of 5 Pa, compared with 100 000 Pa typical atmospheric pressure.