gut feeling is that you shouldn't have high levels in the same multicore as you will have the Mic levels coming the other way
especialy for recording

however
if you must

try running a balanced line cue send to the headphone amp located in the rec studio
...
this may be marginaly better that unbalanced headphone levels and currents in the multicore

multicore should be pair shielded

can you run another multicore for the high levels
or simple speaker leads for post HPamp
seperate to the Mic levels

It's not really a question of impedance, but of signal levels and balancing. Ideally, you need to split the function of headphone mixing and amping/distribution. You should locate the headphone amps in the studio, fed by balanced lines from the control room. If you do this, you should not have problems in using the returns in the same screened multicore as the mic inputs. Your headphone mixer in the control room would therefore need balanced outputs at nominal line levels (+4dBu). Correspondingly, the headphone amplifiers in the studio would need balanced line-level inputs. You have XLRs on your multicore, but the headphone amps and mixer could have either XLRs or TRS 1/4 inch jacks with appropriate local patch cables.

If you need to retain your existing gear with unbalanced outputs, don't try to put the headphone feeds down the multicore - run a separate cable between the control room and the studio. Four-pair with overall screen should be OK for this.

I run light weight speaker cables as HP extenders

7.5 amp twin core flex (round - not fig8)
it is around the same size as a Mic cable and rolls up just as easily
it is also used for my low powered wedges

buy it at the hardware store and can neatly accept both soldered XLs or 6.5mm jacks

Here is the answer to your problems.

It's fine to run your mono/stereo headphones mix, from your mixer/audio interface/auxiliary line level outputs, into your snake.

At the box end, you will then take the 1/4" or XLR outputs from your snake and patch that into your headphone amplifier inputs, which will also require some AC from a wall outlet. This way you're not feeding extremely low impedance, high current signals up the snake, that can inductively couple, causing feedback within the snake. Only line level back fed up the snake to a headphone amplifier, would be the most appropriate way, to do what you want to do.

Don't blow your hearing!
Ms. Remy Ann David

Kev wrote: ... try running a balanced line cue send to the headphone amp located in the rec studio

but I do like the HP control in the Control Room cos I don't trust the punters
... if they bring their own HP's
then thay can do what they like