A uniform difference conjecture for Montgomery's 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 $A$. Conjecture 2. One expects

$$\max_{\frac{1}{2\log T}\le v\le 2\log T}\left|F(xv,T)-F(x,T)\right|\ll\frac{T\mathcal{L}(T)}{\beta(T)^2}$$

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

The paper states that this conjecture implies the preceding conjecture with the same choices of $\mathcal{L}(T)$, $\beta(T)$, and $W(x)$; it is presented as a formulation entirely in terms of $F(x,T)$, but remains unproved.