I am sceptical it will tell you anything about perceived audio quality. I have yet to find any studies that show any significant impact of phase response (or any other time domain behavior, for that matter) on loudspeaker preference in realistic conditions.
However, I am interested in phase response for a different reason: it can help selecting speakers that can be combined together in one system. For example, let's say I want to buy a pair of front speakers and a (different) centre speaker. Ideally, I'd want to use a centre speaker that has a similar phase response to the other speakers, otherwise, they will be out of phase in some parts of the spectrum, and will therefore interfere with each other. That, for me, is a good reason to publish phase data.
I believe that phase changes are subtly audible BUT are often obscured by room reflections. The reflections will produce waveform distortions similar to phase shifts which obscure the underlying information. It is difficult to set up a test to validate this. Headphones are probably the easiest way. if you have the ability to create an unity gain all pass phase shifter you can switch it in and out and hear for yourself. The changes are most audible on transients. They are also audible on massed strings.
One speaker system where this is audible is a set of Linkwitz LX minis with Veledyne subs. This system has been converted to an linear phase device using a DEQX processor and a 4th order Linear phase crossover The drivers were measured with the system elevated so they were centered vertically in a room with a 20' ceiling. The nearest wall was also about 10' away. The calibrated mic was placed 1M on the tweeter axis. From this we generated the speaker correction files using FIR filters. Basically this makes the speaker very close to perfectly flat phase and frequency response at one point in space.
Then the speakers were careful positioned in a large listening room that had been carefully treated using 3 spaced quilts on each sidewalk, a cloud over head and bass traps with diffusion plates in each corner. There is also a very large riser maybe 8x14' that was built that has two low Q limp mass bass traps tuned to the null frequencies of the room. The traps were placed at the location of pressure maxima at those null frequencies. Each device was careful placed and adjusted for maximum results this took over 100 hours of effort. The RT60 is very uniform at around 250mS across the entire spectrum down to 100Hz using an OMNI source.
Room EQ was adjusted using minimum phase parametric EQ at each of the remaining
peaks. The resultant response is flat from 50Hz to 15Khz +/-1.5dB at the MLP.
The listening result was a distinctly better when compared with the same speakers in the same room and position using LR24 at the same frequencies. Levels were closely matched as was the frequency response.
The improvements in clarity and articulation are the most notable.