Homotopy-colimit endomorphism conjecture for full-twist eigenmaps

Let $\operatorname{FT}_n$ be the full-twist complex and let $\alpha:\1\rightarrow\operatorname{FT}_n$ be an eigenmap. Form the directed system

$$\1\longrightarrow\operatorname{FT}_n\longrightarrow\operatorname{FT}_n^{\otimes 2}\longrightarrow\cdots$$

with transition maps induced by tensoring with $\alpha$, and let ${\mathbf L}_\alpha$ be its homotopy colimit. Let ${\mathbf P}_\alpha$ be the corresponding projector, and write $\operatorname{HOM}$ and $\operatorname{END}$ for internal Hom and endomorphism objects. Homotopy-colimit endomorphism conjecture. The homotopy colimit ${\mathbf L}_\alpha$ is isomorphic to the internal endomorphism ring of the corresponding projector. In particular,

$$\operatorname{HOM}(\1,{ \mathbf L}_\alpha)\cong\operatorname{END}({\mathbf P}_\alpha).$$

This is stated conjecturally for any eigenmap; the top and bottom projector cases are described as known examples, but the general assertion remains open.