Preconditioning for Nonsymmetric Singular Matrices?

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I am in a real pickle. I need to solve a linear system $Ax=b$, where A is non-symmetric, singular, and square and of very high dimension. (328x328)

I have tried using multistep splitting iterations per Morikuni (2017; https://arxiv.org/abs/1504.01713), but I have the issue of the iteration matrix not being semi-convergent- and when I apply it to GMRES, I get no solution.

Would really appreciate any references you guys might be able to provide. Thanks!