Prime correlation conjecture for Kloosterman sums and cusp-form eigenvalues

Let $f$ be a fixed primitive cusp form, holomorphic or Maass, and let $\lambda_f(p)$ denote its Hecke eigenvalue at a prime $p$. Write $\mathcal{L}=\log X$. Prime correlation conjecture. For all sufficiently large $X$,

$$\sum_{p\leq X}\lambda_f(p)\mathrm{Kl}(1,p)=o\left(X\mathcal{L}^{-1}\right).$$

The conjecture is proposed as a correlation estimate between Kloosterman sums and Hecke eigenvalues. If true, the source says it would imply that for $100\%$ of primes $p$, one has $\lambda_f(p)\neq\mathrm{Kl}(1,p)$ for each primitive cusp form $f$, giving a negative answer to Katz's modularity problem III. No resolution is given.