I am working on a problem where the following equation came up

$${\bf X}_1{\bf A}{\bf X}_2{\bf A}^T{\bf X}_3{\bf A}-{\bf X}_4={\bf A}$$

where ${\bf A}$ is an arbitrary $n\times n$ and ${\bf X}_i$s are unknown diagonal real matrices. My question is if it is feasible and if there is a computationally tractable way to solve it.