just some thoughts -
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Z Screw already working against gravity, not sure backlash nut has value here. If this was a horizontal Z Screw, then I’m with you.
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Extensive videos online exploring single vs dual geared extruders don’t pan out with ‘slippage’ theory; if anything, single gear extruders lead to less artifacts in the print. dual geared extruders can lead to imperfections in the visual quality of the outer perimeter, because any minor lack of concentricity means you can squeeze out more filament on one part of the turn than another part. When only one gear is at play with an idler, that problem goes away. Check out youtube for several videos illustrating this well.
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Biggest Issue is getting to be perfectly vertical when installed; even off by 1 degree will cause Z Wobble. It is just a major challenge to have two long rods almost 500mm be perfectly vertical with 360 degrees of rotation. Even 0.01mm of lead screw later movement will translate to visible z wobble on the print surface.
TBH, the only real solution I’ve found, unless you invest a lot of money a la CNC, is Wobble Wings. Low cost & incredibly effective. I couldn’t believe the result till I saw it myself on a calibration cube - perfectly flat walls.