The most common objection I have heard in past six months during my job search. Another two objections that I have heard are "too much experience" and "your past salary was too much, we don't want to offer less but can't afford to pay you what you deserve."
You're don't necessarily deserve $n if the market can no longer bare it. There's nothing wrong with taking $(0.8n) if that's what the market has determined your skills are worth.
The problem is that the company is afraid that you'll bolt the second you get an offer at the n level again. They'd rather not hire you at all than hire you, invest in you, and then lose you just as quickly afterward.
Offer a longer term contract then. The guy in the video was at his last job for 15 years. I'm sure he'd be OK with that assuming the benefits were good enough.
The hard part about this is convincing the HR department that you can self-relocate and don't want relo money.