My dear Scout had SLO.
People tend to use the names interchangeably, but they are NOT the same, although they are both auto-immune diseases. SLO is the right name for what you suspect. Greyhounds are one of the breeds that seem to be more prone to it.
Scout's nails were never cured, but we slowly got them under control with an initial bout of tetracycline/supplements, followed by the supplement protocol listed above. His last couple of years he only lost one nail a year, and it only appeared painful for 3-4 days. My memory is dimmed with time, but I don't think we ever had to use prednisone. The affected nails always stayed gnarly. At first they were soft when I cut them. As he got better, they hardened up but were often hollow inside. He hated having his nails cut, but they grew very fast.
As for "giving" your dog SLO with food changes; don't beat yourself up about it. ** happens.
Best of luck to all the nail dogs out there.