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Yes, it's true that e-ink displays do not require power to maintain the image.
As for the difference between sleeping and fully powered off, I'm not sure there is much of one. If I had to guess, sleep mode uses only enough energy to power the RAM allowing the e-reader to wake instantly, and the energy used is likely no more than the natural discharge rate of an idle rechargeable battery. I've also seen the suggestion that any energy saved by powering off is offset by the energy used during the power-on sequence.
Speaking in practical terms, my first generation Kobo Aura used daily with the front light can go for several weeks between charges, and I never fully power it off.
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