The weighted ℓ1−2 minimization has recently attracted some attention due to its capability to deal with highly coherent matrices. Notwithstanding the availability of its stable recovery guarantees, there appear to be some issues not addressed in the literature, which are (i). convergence of the solver for the weighted ℓ1−2 minimization analytically, and (ii). detailed analysis of relevance of general weights to applications. While establishing the convergence of the solver of the weighted ℓ1−2 minimization, we demonstrate the significance of general weights, w∈(0,1), empirically through some applications, including the reconstruction of magnetic resonance images. In particular, we show that the general weights attain significance when we do not have fully accurate or fully corrupt information about the support of the signal to be reconstructed from its linear measurements. We conclude the work by discussing a numerical scheme that chooses the partial support and the weights iteratively. © 2022 Elsevier Inc.