How can I Axtinguish enemy Pyros?

The following is under the trivia section of the Axtinguisher page on the TF2 wiki:

Although Pyros and Demomen using the Chargin' Targe are resistant to afterburn, they can be crit by the Axtinguisher if the hit connects before resistance takes effect.

Now, I know how to inflict critical damage against these opponents with the flare gun: simply hold down M1 (keeping hold of the button), and scroll down.

This works, apparently, because the nature of particle-based weapons mean that fire particles linger for a short period after being fired, during which time the enemy is counted as being "on fire" (I don't have a source for this, but remember reading it somewhere).

However, when I try doing the same thing with the Axtinguisher (using the melee key, as opposed to scrolling down), no inflicted critical hit occurs.

Now, why does this not work for the Axtinguisher? The same conditions are met, so surely I should inflict critical damage (from behind).

More importantly, how can I get this to work?

I've tried it on tr_walkway with stationary Pyros, and the Degreaser-Axtinguisher combo doesn't work.

I have noticed, however, that if an ally continuously blasts an enemy pyro with his flamethrower, and I, during that time, hit the enemy with the Axtinguisher from behind, it will score a critical hit.

Is there any way I can Axtinguish Pyros (and Charge Demoknights) without the help of an ally Pyro?