I have picked up from this site that the Stax 009's are something special and I look forward to hearing them. Their high cost is an issue with 2 kids starting college, but even that can be finessed. However my enthusiasm is dampened by the several comments I have read, including your own (M. Arnaud), about their revealing nature making them unsuitable for use with significant areas of music. This does not sound like a tonal issue which one is going to get used to in any reasonable period of time. While I love classical music and these phones would probably be excellent with them, my tastes are pretty catholic (note the small "c"). So for me this issue takes them out of the realm of "must have" to simply "great phones" but not on my immediate to-do list, rather like the HE-90's.
I also have a residual irritation with Stax about not getting the tonal balance right on its flagship. This has been a recurring issue with Stax, that so many of its otherwise great phones have suffered the defect of not being listenable on a large body of music.
My first Stax was the SRXIII, which is a fine phone, but somewhat thin sounding, not so much because of a treble peak but because of deficient bass. Many years later I got the contemporaneous Stax SR5, which I think has the tonal balance issue down much better and I found myself thinking, I wish these had been my first set rather than the SRXIII.
Similarly, all the Lambdas I have had and heard have suffered from treble peakiness to one degree or another. However their drivers work well in the Sigma enclosures, for which, of course, they were originally designed. While the Sigmas may have too little treble and tend towards some boominess, still, I like their sound and can be pretty sure if I give them to non-audiophiles they will appreciate these phones without handing them back to me saying they are harsh sounding. Similarly, the 003's although somewhat limited in resolution, are very listenable on a wide variety of music and I have enjoyed them considerably even on the little Stax portable amp.
Now I am not sure if acoustic detail can be fully resolved without good high frequency performance of the driver. So maybe inevitably any highly resolving phone will be treble rich compared to less resolving phones. But the comparison of the Lambdas with the Sigmas shows that you can take identical drivers and still produce a much less harsh sounding phone with a different enclosure.
Also, all things being equal, pumping up the treble will increase high frequency detail where the upper harmonics are, thus making the sound more detailed, but unnaturally so. I think that this is a ploy with some audiophile phones to give them a resolution bump.
I have also felt that given the treble issues in many Stax phones, the company should have at least included a defeatable tone control or filter on its amps. But of course this type of control gets attacked by audio purists.
I believe that one of the fundamental problems with high-end audio is that is too much of a male-dominated field. The reason this is a problem has nothing to do with politics but is that men tend to have poorer high frequency hearing even when quite young and it lose it even more than women as they age. The difference seems to be in part inherent in the basic physics of sound such as the resonances of the auditory canal and ossicles (the bones that transmit sound from the eardrum to the cochlea) are higher in women and children. Also men tend to expose themselves to more loud noises and lose hearing for that reason as well as other health reasons.
So when a middle-age man develops a phone, it may sound good to him but treblish to a woman or younger person or even older man who has kept his hearing in better shape. I think this hearing issue has shaped the audio business to its detriment.
These are some sources I googled up, not necessarily the best. Interestingly there is speculation that the high frequency hearing differences between the sexes may be responsible for the well-documented female advantages in learning language skills, i.e. their better high frequency hearing makes it easier for them to understand what is being said to them as children.
http://www.genderdifferences.org/hearing.htm
http://www.arlenetaylor.org/sensory-preference/423-gender-hearing-differences