Hurwitz's explicit class-number-sum conjecture modulo 7

For a prime $p$, integers $a$ and $n\ne 0$, define

$$H_{a,p}(n):=\sum_{\substack{|m|\leq 2\sqrt{n}\\ m\equiv a\pmod{p}}} H\left(4n-m^2\right).$$

Hurwitz's conjecture. For a prime $\ell$ and $a\in\mathbb{Z}$,

$$H_{a,7}(\ell)=\begin{cases}\frac{\ell+1}{3}&\text{if }a\equiv\pm1\pmod{7},\text{ and }\ell\equiv1\pmod{7},\\frac{\ell+1}{4}&\text{if }a\equiv\pm1\pmod{7},\text{ and }\ell\equiv3,6\pmod{7},\\frac{\ell+1}{4}&\text{if }a\equiv\pm2\pmod{7},\text{ and }\ell\equiv3,5\pmod{7},\\frac{\ell+1}{4}&\text{if }a\equiv\pm3\pmod{7},\text{ and }\ell\equiv5,6\pmod{7}.\end{cases}$$

The supplied text presents this as a conjecture attributed to Hurwitz; it does not include a proof or resolution of the explicit formula.