Jutila's conjecture on moments of quadratic character sums

Let $S(X)$ denote the set of all non-principal quadratic Dirichlet characters of modulus at most $X$. For $k=1,2,\dots$, and $X\geq 3$, $Y\geq 1$, define

$$S_k(X,Y):=\sum_{\chi\in S(X)}\left\lvert\sum_{n\leq Y}\chi(n)\right\rvert^{2k}.$$

Jutila's conjecture. For all $k=1,2,\dots$ and $X\geq 3$, $Y\geq 1$, there are coefficients $c_1(k)$ and $c_2(k)$ depending only on $k$ such that

$$S_k(X,Y)\leq c_1(k)XY^k(\log X)^{c_2(k)}.$$

This conjecture concerns higher moments of quadratic Dirichlet character sums. The paper studies these moments in a restricted range and obtains asymptotic results and a lower bound for the exponent of the logarithmic factor, but the stated estimate is not resolved here.