Kloosterman-fraction trilinear estimate conjecture

Let

$$S_{A,M,N}:=\sum_a\sum_{(m,n)=1}\nu_a\alpha_m\beta_n\operatorname{e}\mathopen{}\mathclose\bgroup\originalleft(\frac{a\overline{m}}{n}\aftergroup\egroup\originalright),$$

where $M\leq m<2M$, $N\leq n<2N$, $A\leq a<2A$, and $\|\cdot\|$ and $\|\cdot\|_\infty$ denote the $L_2$ and $L_\infty$ norms. The conjecture concerns the range $A\ll (NM)^{\frac12+\varepsilon}$. Kloosterman-fraction trilinear estimate conjecture. In that range,

$$S_{A,M,N}\ll \|\alpha\|\|\beta\|\|\nu\|(M+N)^{\frac12+\varepsilon}+\|\nu\|A^{\frac12}\mathopen{}\mathclose\bgroup\originalleft(\|\alpha\|_\infty\|\beta\|N^{\frac12+\varepsilon}+\|\alpha\|\|\beta\|_\infty M^{\frac12+\varepsilon}\aftergroup\egroup\originalright).$$

Such an estimate would improve the admissible length of the Dirichlet polynomial in the zeta-function mean-square problem; currently available estimates give weaker ranges, and the conjectured bound is not known in general.