The $(S_2)$-condition conjecture for edge rings of triangular cactus graphs

Let $G$ be a triangular cactus graph, meaning a cactus graph whose blocks are all $3$-cycles, and let its diameter be the maximum distance between two vertices of $G$. The $(S_2)$-condition conjecture. The edge ring associated to a triangular cactus graph of diameter at least $4$ satisfies Serre's condition $(S_2)$. This conjecture extends the known cases of triangular cactus graphs of diameter at most $3$, whose edge rings are normal and Cohen--Macaulay; the paper proves the claim for triangular cactus graphs of diameter $4$, while the general case remains open.