Hurwitz's linearity conjecture for class-number sums modulo 5

For a prime $p$, integers $a$ and $n e 0$, 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 every $a,L\in\mathbb{Z}$, there exist constants $c_1,c_2\in\mathbb{Q}$, given explicitly by the following cases, such that for every prime $\ell\equiv L\pmod{5}$,

$$H_{a,5}(\ell)=\begin{cases}\frac{\ell+1}{2}&\text{if }a\equiv 0\pmod{5},\text{ and } \ell \equiv 1\pmod{5},\\frac{\ell+1}{3}&\text{if }a\equiv 0\pmod{5},\text{ and } \ell \equiv 2,3\pmod{5},\\frac{\ell+1}{3}&\text{if }a\equiv \pm 1\pmod{5},\text{ and } \ell \equiv 1,2\pmod{5},\\frac{5\ell+5}{12}&\text{if }a\equiv \pm 1\pmod{5},\text{ and } \ell \equiv 4\pmod{5},\\frac{5\ell-7}{12}&\text{if }a\equiv \pm 2\pmod{5},\text{ and } \ell \equiv 1\pmod{5},\\frac{\ell+1}{3}&\text{if }a\equiv \pm 2\pmod{5},\text{ and } \ell \equiv 3,4\pmod{5}.\end{cases}$$

The paper proves this conjecture using identities for mixed mock modular forms, so it is solved.