The statement is far to general to be true in any significant number of real-world cases.
And this is why:
"Stability" means the amplifier doesn't become unstable, breaking into oscillation or creating some objectionable degree of distortion.
If the load is within an amplifier's design parameters, the amplifier should be stable under all possible conditions. All serious amplifier designers have considered the load range very early, because it directly affects the cost of manufacture. If not, the amplifier is poorly designed, or the load is inappropriately applied outside of design parameters. Generally, headphones are not a difficult load to drive as compared with speakers because though impedances may dip, the applied power is microscopic compared with the demands of a speaker. Most headphone/amp mismatches relate to available voltage, with higher impedance headphones requiring more voltage to achieve the required power, and certain types, like IEMS, require very little voltage. Power = V squared over R, so holding total efficiency the same, if you raise R (resistance, or simplified impedance) you must also raise V (voltage) to get the same power and resulting volume. Required power relates to headphone type efficiency, which doesn't change an awful lot within general headphone type categories. Different categories, like low Z IEMs vs high Z full size open, do have notably different requirements.
Because different types of headphones have vastly different requirements, we end up with some combinations that work better or worse, and amps with switches to compensate for different headphone categories. But amplifier stability should not be an issue unless you have a horrible load combination on a really badly designed amp.