Finiteness conjecture for spectral lifts of powers of Hogancamp's endomorphisms

For each $n\geq 2$, let $P_n$ be the categorified Jones–Wenzl projector and let $U_n\colon \mathrm{q}^{2n}\Sigma^{2n-2}P_n\to P_n$ be Hogancamp's chain map. For each power $U_n^k$, let $\beta_{n,k}$ denote its obstruction class to lifting to a map of spectra. Finiteness conjecture. For each $n\geq 2$, there exists some $k>0$ for which

$$\beta_{n,k}\simeq 0.$$

Equivalently, some power $U_n^k$ lifts to a map of spectra $\mathcal{P}_n\to\mathcal{P}_n$. The conjecture is verified in the paper for $n=2,3$; existence of such lifts does not by itself guarantee that they are well-defined.