A conjecture in [MATCH Commun. Math. Comput. Chem. 89 (2023) 513-530] states that for any unicyclic graph \( G \) of order \( n \ge 3 \), the spectral radius of any degree-based matrix (i.e., a matrix whose entries are functions of vertex degrees) lies between \( \rho(C_n) \) and \( \rho(S_3^+) \), where \( S_3^+ \) denotes the unicyclic graph obtained by attaching \( n-3 \) pendent vertices to a single vertex of a triangle. We disprove the upper bound for the second Zagreb matrix. Exhaustive computations for \( n=5 \) to \( 11 \) show that the true maximizer is a triangle: for \( n=6 \) it has one leaf on each vertex \( (1,1,1) \); for all other \( n\ge5 \) it has leaf counts \( (\lfloor\frac{n-3}{2}\rfloor,\lceil\frac{n-3}{2}\rceil,0) \) (i.e., one cycle vertex receives no pendent leaf). We conjecture that these graphs uniquely maximize the second Zagreb spectral radius for every \( n\ge5 \).