Does Power Cord Require Burn-In To Sound Good?

It is not a matter of "So says you". Simple fact, that so many of his claims are wrong or unsubstantiated. He should stop his rants long enough to realize that AD is Analog Devices and still very much in business. BB and LTC were just purchased by larger competitors and the the sub is often better than the pieces.

How power influences the sound depends upon the equipment: The power supply in an amplifier could do so thorough a job of filtering out noise along with hum that it might make power conditioning irrelevant. If you use single ended triode amplification it will always draw the same current rather than demand more current for loud passages in a recording and the resistance of a less expensive power cord will be trivial. Higher audio frequencies are easier to filter out than 60 Hz hum and many vacuum tubes have cutoff frequencies at higher radio frequencies.
Watch out for the placebo effect for cables in general. There is some junk science used to market audio.

Be suspicious when the recommended break in period is likely to take longer than the return policy last. Pretty hard to get 200 hours of listening in 30 days for most people. Just a thought. 

Like all audio products, the power cord must burn in for 90 days, i.e. until the credit cardfcannot be refunded.  Ask any sales person who works on commission.