It appears that to get to the shiny trinket, I'll have to untie that Gordian knot, but I don't have any idea about how I could achieve that.
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The correct way is pretty straight forward:
The tricky part is to get from the lower left to the lower right spawn point on the "Gordian Knot" screen, but with the right timing you can just press
right on lower left span point and run through over the moving platforms onto the right ledge.
On the way back you have to pass that place again, and then you can't just run through. You have to stand on the edge of the moving panel where the spikes can't hit you and jump to the ledge at just the right moment.
It turns out there is another way. With frame perfect timing you can actually go there straight from the Gordian Knot room without passing through Backsliders first.
See 5:55 in this youtube video: