Thought we had covered all the scenarios but overlooked what may have been a fairly plausible one. Tex Colbert, CEO of Chrysler determines that '57 Imperial, despite its all new styling, will not be strong enough as a brand to take on Cadillac by itself. So he acquires Packard in April, 1956 and tools a '58 that mates Imperial body to longer wheelbase forward of firewall. It is determined that Request styling best blends with Imperial body. Imperial's fins get clipped on the Packard.
Because Chrysler owns Conner Ave. and will get screwed when S-P breaks lease early, Colbert decides to work deal whereby '57 Imperial and '58 Imperial-based Packard get built there. (Imperial production moved from Jefferson Ave. where it was built alongside Chrysler, to its own plant on Warren Ave. for 1959 to improve quality, only to be moved back to Jefferson for 1962 due to low volume).
To position Packard above Cadillac (and Imperial), Colbert also buys Utica V8 facility from S-P for a song, has Packard's engineers finish development of 480 CID V12 and builds it there. Torsion-Level is also worked in (whereas all '57 Chrysler Corp. vehicles came with torsion front only). Imperial gets Packard's T-L too.