The diagonal magnitude homology conjecture for triangle attachments to a square

Let $C_3$ and $C_4$ denote cycle graphs of lengths three and four, respectively. Form a graph by gluing any number of copies of $C_3$ along single edges to a single copy of $C_4$, and assume that no pair of triangles is glued to opposite faces of the $4$-cycle. General triangle-square gluing conjecture. The resulting graph has diagonal magnitude homology. The claim extends the two-triangle case and is motivated by computations in the source; its general case remains open.