Very interesting question indeed. Let's try to see what are the things at play, when a speaker is being driven by an amplifier.
Firstly, lets be clear that the sensitivity mentioned is the Sound Pressure Level it produces at a distance of 1 metre by 1 watt. Then onwards, any increase of 3dB demands double the power. So, a 86dB speaker would need 1 watt to produce 86dB of SPL at 1 metre, it would need 2 watts to produce (86 + 3) 89 dB, 4 watts fot 92 dB and so on. But this has a caveat, this is true assuming the impedance remains constant throughout the SPL and frequency.
Next we need to know that there are broadly, three types of electrical loads, Resistive, Inductive and Capacitative. A speaker is a combination of all three of these and together they define the impedance of a speaker. Since these load characteristics change with frequency, we have a far more complex behaviour of a speakers electrical character when we take the crossover and speaker cables into the total LCR Circuit.
Keeping the above points in mind, we can say that whether a speaker of 86dB will require either more than or less than 2 watts to produce 89dB of SPL depends on what frequency it's going to reproduce at that instant. That's the reason why the instantaneous power delivery depends on the source material too.
Let's come to the cabinet part now. Imagine trying to force a driver to stop while its making sound. Any external damping applied to the driver would change it's impedance characteristics and produce back EMF. A smaller cabinet would contain less volume of air in it. So the compressibility resistance of the air is higher and hence the driver damping is more. Incase of a larger cabinet the compression factor is logarithmically less and if used with 2nd order or higher crossover then the phase differential compression from different drivers would make the damping less. That would result in different impedance curve based on cabinet volume, and hence the amplifier's drivability comes into picture.
Along the impedance curve of a speaker there are regions where "voltage leads current" and "current leads voltage". These role reversals take a lot of additional "pressure" on the power supply of the amplifier. If the AVR is sufficiently powerful to handle these then I don't see any theoretical issue of not being able to drive slightly inefficient speakers.
Please keep in mind that to indentify the compression artefacts of an underpowered speaker is extremely difficult, atleast just by listening.