On a countable system of convolution equations

Let $(\varphi_1(z),\dots,\varphi_m(z),\dots)$ be a countable collection of entire functions $\varphi_j(z)$ of order 1 and minimal type, and let the estimate
$$ \sum_{j=1}^{\infty}|\varphi_j(z)|^2\le a(\varepsilon)\exp\{\varepsilon|z|\}, \quad z\in\mathbb{C}^n, $$
hold for every $\varepsilon>1$. The countable system of nonhomogeneous convolution equations
$$ M_{\varphi_j}[y]=g_j(z), \quad j\ge1, $$
is studied, where $M_{\varphi_j}$ is the convolution operator, with characteristic function $\varphi_j(z)$, acting on the space of holomorphic functions over some convex domain $\mathcal{D}$. Necessary and sufficient conditions for solvability in the space $H(\mathcal{D})$, as well as uniqueness for a solution to the system, are established.