Does this represent anything more than a single isolated case?
Is the small difference shown audible?
Amps used? Two different amps or single amp with both functions?
Same headphones but with different cables or two different headphones?
Single trial or 10 trials?
What controls were used?
Does HeadRoom sell balanced cables?
I think the two frequency responses are very close. Though, usually the balanced amps are superior. e.g. Schiit Mjolnir vs Asgard
I don't think only balanced cables can make audible difference.