Roy Smith wrote

Why the staggered shutdown? Waiting to hit certain weight
limits as fuel burned off?

YEP!

Once you had shut down two engines, what would happen if one of
the remaining two failed? Was the airplane capable of flying on
one?

At least until we could get another started. That was the reason
for the 500' limitation on two engines. Wasn't adhered to very
often though. As I recall, it only took about 10-15 seconds get
one going again.

This fuel saving method only works on constant speed turboprop
engines. The Allison turned the prop a constant 1061 rpm....
TO...cruise...and... landing. The compressor section required
about 6000hp at that constant speed regardless of the power being
delivered to the prop (up to 4600hp).

Bob Moore