Hurwitz's partial linearity conjecture for class-number sums 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. If $L=3,5,6$ and $a\in\mathbb{Z}$, or if $(a,L)\equiv(\pm1,1)\pmod{7}$, then there exist constants $c_1,c_2\in\mathbb{Q}$ such that for every prime $\ell\equiv L\pmod{7}$,

$$H_{a,7}(\ell)=c_1\ell+c_2.$$

This conjecture is stated as a computer-data-based prediction in the source, and the supplied text does not establish a resolution for it.