  We show that if the probabilistic logarithmic-space solver or the
deterministic nearly logarithmic-space solver for undirected Laplacian matrices
can be extended to solve slightly larger subclasses of linear systems, then
they can be use to solve all linear systems with similar space complexity.
Previously Kyng and Zhang proved similar results in the time complexity setting
using reductions between approximate solvers. We prove that their reductions
can be implemented using constant-depth, polynomial-size threshold circuits.
