Hooley's Hypothesis $K^*$ for diagonal Waring representations

Let $r_{k,k}(l)$ denote the number of representations of a positive integer $l$ as a sum of $k$ positive $k$th powers. Hooley's Hypothesis $K^*$. For $k=2$ and every $\epsilon>0$,

$$\sum_{l=1}^{N}r_{k,k}(l)^2=O\left(N^{1+\epsilon}\right)\qquad(N\to\infty).$$

For $k\geq3$,

$$\sum_{l=1}^{N}r_{k,k}(l)^2=O(N)\qquad(N\to\infty).$$

This is an average-value analogue of Hardy and Littlewood's Hypothesis $K$, which is false for $k=3$ and expected to be false for larger $k$. The conjecture remains open, and would give important mean-value estimates for Weyl sums.