Sparrow can slide down the rope with both hands bound together even though both ends are fixed as you reason, because he is able to pass the rope through his wrist joints as one of the cursed...since this scene is before the revelation in the cave when moonlight reveals the fact that he is one of them. I you watch carefully, frame by frame you will see this happen and also see that Sparrow's face reacts to effort and mechanical discomfort of the action prior to the death slide.
max
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