Among different migration algorithms, least-squares reverse-time migration is the preferred choice for quantitative seismic imaging. The applicability of such scheme depends on the derivation of proper pre-conditioners. In the context of extended domain in a pure acoustic media, the pseudo-inverse Born operator is the recommended pre-conditioner, providing quantitative results within a few iterations, but limited to purely velocity variations. Recently, an efficient weighted least-squares approach has been proposed to extend the applicability of the pseudo-inverse Born operator in the presence of density variations. As expected in the case of multi-parameter imaging, the results using this method suffer from crosstalk artifacts. In order to mitigate this issue, we present variable density pseudo-inverse Born operator constrained with $\ell_1$-norm for each model parameter. The fast iterative shrinkage-thresholding algorithm is used to solve the optimization problem. This iterative scheme is based on soft-thresholding method where no wave-based operators are involved. Numerical tests are used to demonstrate the robustness of the proposed method against crosstalk artifacts and sparse shot acquisition geometry.