A uniform upper-bound conjecture for the smoothed pair correlation function

For $T\ge 3$, let $\mathcal{L}(T)$ and $\beta=\beta(T)$ be non-decreasing continuous functions satisfying

$$1\le \beta(T)\ll \log^3T,\qquad \log T\ll\mathcal{L}(T)\ll\beta(T)\log^2T.$$

Let $W=W(x)$ be increasing with $W(x)\gg(\log x)^A$ for a sufficiently large constant $A$. Let $F_\beta(x,T)$ denote the weighted pair correlation function defined in the paper. Conjecture 1. One expects

$$F_\beta(x,T)\ll T\mathcal{L}(T)$$

uniformly for $W(x)\ll T\ll x^{1/2}\log^2x$.

Any improvement over the trivial bound in this range would improve the Riemann-Hypothesis bound for the prime number theorem error term; the source says there is presently no hope of proving these conjectural estimates.