With bleed valves shut and whether packs are on or off, the piping will still retain a little of the contaminated air and send it through when the valves are reopened after start.
I agree that tailwind components are the most likely cause of the offending ingestion A surge from the fiery end has been known to travel upstream to the HP bleed stages and beyond.
Blah blah blah.....
Before jet engines were trusted with supplying clean cabin air, turbo-compressors were used for cabin air back in the days of the 707. The arrangement was a turbine fed by engine bleed air connected to a compressor with its own fresh air intake for the cabin.