The Black Hole......

Before you can make claims about the ferrite over the headphone cable you need to address some real "confusors" to make sure you are actually addressing what you think you are.

1) On the phone was it in air plane mode? Phove have 5 or more radios and anything attached to the phone beomes part of its antennal structure and will influence those radios which in turn will affect all manner of other internal processes. Just turn them off.
2) Uncompressed, Flac or lossy files. The Lossy fines may not be identical on every playback. Flac will use a little more processor but should be pretty consistent. An uncompressed file will be the most consistent.

Apple decreed that any "made for iPhone" headphone would need separate ground returns to the minijack 10? years ago. It had been pretty common since that made for iphone logo was essential to sell headphones.

Headphones, being low impedance, usually only have the mike cable shielded. The other wires can be as small as 32 AWG stranded (smaller in some cases). Its not for cost since the wire gets more expensive when its finer gauge but for flexibility. Again more important to sales than sound.

Re radios above, the ferrite could well tune the cable to resonate with one of the radios or interact with incipient instability on the headphone amp in the phone. Not likely but possible.
 
Before you can make claims about the ferrite over the headphone cable you need to address some real "confusors" to make sure you are actually addressing what you think you are.
The phone was playing various stuff over wifi or with stored wav files with Airplane Mode turned on.
Below is AC adaptor laptop laptop with WiFi on, laptop headphone out driving the headphones placed on sofa chair and close recording using Logitech 16bit USB microphone on mic stand, without ferrite and with ferrite at headphone end of cable.
Normal
Ferrite
 
Getting bored? Why don't you build something? We have a few folks interested in modding dacs again, maybe for so long as lockdown lasts.

I've had some loose thoughts of building something, mostly around a "cement half A to half B" effort between one of TI's DSP amps and a tube amp. If the little output inductors have hysteresis distortion, just think what replacing those with a pair of 6CA7s / gigantic audio power output transformers would do!

Turning a pair of out-of-phase quiescent 50% PWM outputs at 100's of kHz into a, say, 24 kHz -3db low-pass bandwidth analog signal by simply using an RC should be no problem. Grid drive current for the 6CA7s should be no problem.

My question is why would I want to "lampizate" an amplifier capable of driving speakers directly, using the sonically worst part of a P-P tube amp - the power output stage? At the moment, unsure if a feedback path from output transformer secondary is even possible... Who knows, it might sound good somehow running open loop, using all the available DSP to "fix up" the open loop response of the tube amp output stage, but - for me - such an effort is well into the realm of having nothing better to do.

Wireless tube guitar amp - BT connection from guitar, tone controls in DSP, tubes to handle the glow and transient punch!

I'd rather see if I can improve the sound of the TI amp itself, that's the "DAC" I'd be most interested in modding.
 
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The phone was playing various stuff over wifi or with stored wav files with Airplane Mode turned on.
Below is AC adaptor laptop laptop *with WiFi on*, laptop headphone out driving the headphones placed on sofa chair and close recording using Logitech 16bit USB microphone on mic stand, without ferrite and with ferrite at headphone end of cable.
Normal
Ferrite

As you just stated above "with wifi on"
 
2) Uncompressed, Flac or lossy files. The Lossy fines may not be identical on every playback. Flac will use a little more processor but should be pretty consistent. An uncompressed file will be the most consistent.
Please elaborate why there should be any difference in the analog delivery of files of same format and also files of different formats and what those differences might be ?.
Thanks in advance.
 
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Max, yesterday I accidentally found the 'root' of our questions, answered, by Richard Heyser several decades ago. I knew him well, but nobody likes to believe what he told me 50 years ago, but here are his own words: Just Google 'Richard Heyser'. Then download "Is Heyser Still Relevant?" from Douglas R. Jones from Columbia College Chicago.
You will find the heart of why we hear things, yet can't measure them (yet). Check it out, along with the other articles from Jones. Good luck!