The nonvanishing and square-root bound conjecture for modular-symbol Ramanujan sums

Let $R_j^+(n)$ denote the scalar quantity associated with the embedded spectral contribution, and let $r(X)$ be the corresponding error term in the twisted Ramanujan-sum asymptotic. Modular-symbol Ramanujan-sum conjecture. (1) For each fixed $j>0$, there exists a nonzero integer $n$ such that $R_j^+(n)\neq0$. (2) For every nonzero integer $n$, every $k>6$, and every $\varepsilon>0$,

$$r(X)=O_{n,k,\varepsilon}\left(X^{1/2+\varepsilon}\right).$$

The first assertion would prevent all embedded-eigenvalue contributions from vanishing simultaneously, while the second predicts square-root cancellation up to $X^\varepsilon$; both remain conjectural in the supplied source.