Quartic large sieve upper bound

Let $\Xi_4(A,B)$ denote the quartic large sieve quantity for parameters $A,B\geq 1$, and let $\varepsilon>0$. Quartic large sieve upper-bound conjecture. For any $A,B\geq 1$,

$$\Xi_4(A,B) \ll_{\varepsilon} (AB)^{\varepsilon}(A+B+A^{3/4}B^{1/2}+A^{1/2}B^{3/4}).$$

The conjecture proposes an essentially tight upper bound for the quartic large sieve, improving on the currently known bound $\Xi_4(A,B)\ll_{\varepsilon}(AB)^{\varepsilon}(A+B+(AB)^{2/3})$. The preceding theorem establishes a lower bound of the same shape up to the factor $(AB)^{\pm\varepsilon}$, but the matching upper bound remains open.