Cohen's modularity conjecture for class-number generating series

Let

$$S_4^1(\tau,X):=\sum\limits_{\substack{n=0\\ n\text{ odd}}}^\infty \left[\sum_{\substack{s\in\mathbb{Z}\\ s^2\leq n}}\frac{H(n-s^2)}{1-2sX+nX^2}+\sum\limits_{k=0}^\infty \lambda_{2k+1}(n)X^{2k}\right]q^n,$$

where $q=e^{2\pi i\tau}$ with $\operatorname{Im}(\tau)>0$, $H(n)$ is the Hurwitz class number, and $\lambda_{2k+1}(n)$ denotes the coefficients occurring in Cohen's formal power series. For each nonnegative integer $\ell$, consider the coefficient of $X^\ell$ in this formal power series. Cohen's conjecture. The coefficient of $X^\ell$ is a holomorphic modular form of weight $\ell+2$ on $\Gamma_0(4)$. Cohen's conjecture connects the class-number generating series with holomorphic modular forms and extends the modularity phenomena known for the generalized class-number series. Its resolution status is not established by the supplied source context.