Lithium conversion e4- won’t run

No. Ignore that white wire.

Org wire at contactor should have B+ at KeyON but you need to have your other meter probe on B- for ref.

Contactor/relay will not close until ready to go.
Controller has many safety checks that must pass, then when a slight pedal press is detected it will close.

Are there any error codes displayed on screen?
Is brake icon illuminated? (release brake)
Is plug icon cleared?
(Post a pic of dash display)