Yes, I believe it does. If you do the kind of fibre-bundle argument Hatcher does at the end of the Smale Conjecture paper you convert the problem of studying Emb(S^1,D^2) into studying the homotopy-type of the space of proper embeddings of an interval into D^2 (with fixed endpoints). You then use the fibre bundle Diff(D^2) –> Emb(I, D^2 rel endpoints), and show the fibre has the homotopy-type of a product of two copies of Diff(D^2).
It’s not clear how to use the fact that you’re on the unknot component to your advantage. Finding critical points of these energy functionals isn’t so easy. I’m probably going to give a related project to an undergrad — finding critical points of this functional, not on the unknot space, but on the space of a “big” satellite knot. There’s at least enough knowledge of the homotopy-type to get a sense for how to `cast the net’.

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