variables $(\xi_{k,1}, \xi_{k,2}), \zeta_k$, and assume ${\mathbf}{Y}_{N,1}, \ldots, {\mathbf}{Y}_{N,m}$ are independent. For each $1 \leq k \leq m$, let ${\mathbf}{A}_{N,k}$ be a $N \times N$ deterministic matrix, and assume $$\label{eq:Aassump}
    \max_{1 \leq k \leq m} \operatorname{rank}({\mathbf}{A}_{N,k}) = O(N^{1-{\varepsilon}}) \quad \text{and} \quad \sup_{N \geq 1} \max_{1 \leq k \leq m} \frac{1}{N^2} \| {\mathbf}{A}_{N,k} \|_2 < \infty,$$ for some ${\varepsilon}> 0$. Then the ESD of the product $$\label{def:PN}
    {\mathbf}{P}_N := N^{-m/2} \prod_{k=1}^m ( {\mathbf}{Y}_{N,k} + {\mathbf}{A}_{N,k})$$ converges almost surely to $F_m$ (with density given by ) as $N \to \infty$.

We conjecture that items and from Assumption \[ass:atom\] are not required for Theorem \[thm:main\] to hold. Indeed, in view of Theorem \[thm:tvcirc\] and [@NgO], it is natural to conjecture that $\xi_1, \xi_2$ need only have two finite moments. Also, our proof of Theorem \[thm:main\] can almost be completed under the assumption that $-1 < \rho_k \leq 1$. We only require that $\rho_k \neq 1$ in Section \[sec:singularvalue\] in order to control the least singular value of matrices of the form ${\mathbf}{Y}_{N,k} + {\mathbf}{F}_{N,k}$, where ${\mathbf}{F}_{N,k}$ is a deterministic matrix whose entries are bounded by $N^{\alpha}$, for some $\alpha > 0$. See Remark \[rem:lsv\] and Theorem \[thm:prodWigner\] below for further details.

Among other 