but it seems strange that the inboards would lift while the outers don't
They don't! The inboards are mounted on the stiffest part of the wing and their in-flight deflection is peanuts. The outer wing bends upwards and takes the outer engines with it, but that is not the direct cause of the twist.
On a swept wing the lift of the outer bits lies aft of the flexural axis of the inboard bits. Consequently the outer wing exerts a torque on the inner bits tending to produce nose down twist. The inboard half (say) of the wing is so stiff that this twist is pretty small, but as you move outboard the bending and torsional stiffnesses of the wing get smaller and more liable to bending and twisting deflections. Adding winglets makes it worse.