This method converges unconditionally for any input with the required sparsity when using numerical representations with a fixed number of bits. No traditional sparse iterative solver has that property without further conditioning the matrix or increasing the run-time to handle the higher precision. As they lay out in the paper (third paragraph of introduction section on the preprint), for certain (rare in practice) matrices direct solvers still offered better runtime if convergence is demanded for all inputs with a certain sparsity.

This article (not the paper) definitely oversells the immediate practical applications; nobody is going to throw out their CG solver and pick this method up in practice to guarantee convergence in this manner.