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$k$-Disk algebras versus $E_k$ algebras

Still, rotations are homotopic to the identity, so it seems (to me) safe to assume that the operads are equivalent as $(\infty,1)$-categories. Am I mistaken? … Also, according to Ginot's notes (page 27, Example 12), the $(\infty,1)$-categories of algebras of $\text{Disk}_k^{fr}$ and $\square^k$ are equivalent, and this leads me to believe that the operads themselves …