The square-mean conjecture for cusp-form coefficients at Piatetski–Shapiro primes

Let $f$ be a cusp form for the full modular group, let $1<c<8/7$, and let $\mathbbm{P}$ denote the set of primes. Write $\lambda_f(n)$ for the normalized Fourier coefficients of $f$. Under the assumptions of the paper's main theorem, there is a constant $c_f>0$ such that

$$\sum_{\substack{n\leq N\\ [n^c]\in\mathbbm{P}}}\left|\lambda_f\left([n^c]\right)\right|^2\sim c_f\frac{N}{c\log N}\quad\text{as }N\to\infty.$$

This conjecture predicts the analogue, for Piatetski–Shapiro primes, of the known square-mean asymptotic over all primes. The source presents it as an unproved conjecture and gives no resolution.