The Ratios Conjecture for local averages of three-cube sums

Fix a real $M\geqslant 0$. Let $\bm{a}\in \mathcal{O}^6$ and $d\in \mathcal{O}^+$ with $\lvert \bm{a}\rvert,\lvert d\rvert\leqslant \widehat M$. Let $\bm{b}\in K_\infty^6$ with $\lvert \bm{b}\rvert\leqslant 1$. Let $A_{F,1}^{\bm{a},d}(s)$ be the Euler product defined from the local averages $\bar{\mu}_{F,1}^{\bm{a},d}(r)$, and write $s=\beta+\sigma(Z)+i\tau$. The Ratios Conjecture. There exists $\beta=\beta_M(Z)\in[0,1]$ such that, uniformly over $Z\in\mathbb{N}$ and $\tau\in\mathbb{R}$,

$$\sum_{\substack{\bm{c}\in \mathcal{S}_1 \\ \bm{c}\equiv\bm{a}\bmod{d} \\ \lvert\bm{c}-t^Z\bm{b}\rvert\leqslant\widehat Z/\widehat M}}\Phi^{\bm{c},1}(s)=\sum_{\substack{\bm{c}\in \mathcal{S}_1 \\ \bm{c}\equiv\bm{a}\bmod{d} \\ \lvert\bm{c}-t^Z\bm{b}\rvert\leqslant\widehat Z/\widehat M}}(1+O(g\widehat Z^{-3\beta}))\,A_{F,1}^{\bm{a},d}(s),$$

for some function $g=g_M(Z)\to0$ as $Z\to\infty$. This is the paper's Ratios Conjecture prediction, expressing the averaged ratio through the leading Euler-product constant; the stated parser status is unknown, so its resolution is not established by the supplied material.