Correct, nobody knows exactly which grease to use…
Anecdote from Germany:
A good friend showed the Sturmey Archer document to his workstore and forced them this way to change the grease of his drive.
Of course they didn’t read it before and were surprised, but they managed the task.
Unfortunately it’s not 100% clear which lube they used - my friend was told “common bike grease”.
…and there is one more problem; the drive has to be disassembled before greasing because there is no refill opening present.
Though I’d not recommend to open the drive without knowing before exactly what/how to do.