Columns of liquid on top of gas, when and why does gas bubble up through the liquid?

I have a long straw ("small" diameter tube of constant cross-section) hanging vertically above me. The straw is filled with liquid (water). I blow a finite volume of air (gas) into the lower end of the straw. The input of gas into the lower end of the straw is such a way that the gas spans the entire diameter of the straw. The gas lifts the liquid column above it, but does not "bubble up through" the liquid. The gas and liquid are now in hydrostatic equilibrium; a column of gas at the bottom, with a liquid column sitting on top of the gas.

I now imagine a "funnel" or "cone" shaped straw, i.e., one where the diameter of the straw is "large" at the top $$(D_{top}>>D_{bottom})$$ and gradually gets smaller towards the bottom $$(\Delta D/\Delta L<<1)$$, where the diameter of this straw at the bottom is same as the previous scenario. Following the same scenario as previously described, I input air into the lower (smaller) end of the straw. I would imagine that as I put a little bit of gas volume in, the entire liquid column would be displaced upwards, and the gas would remain under the liquid column. But I would also imagine that if I put enough air in, enough so that the top of the gas column reaches a certain (larger) diameter of the straw, the gas at this interface will break away as a bubble and rise through the liquid column.

Is this intuition correct? If "yes" what are the underlying physics? What determines if the gas bubbles up through the straw or not?