The complementary second Zagreb index of a graph \( G \) is defined as \( cM_2(G)=\sum_{uv\in E(G)}|(d_u(G))^2-(d_v(G))^2| \), where \( d_u(G) \) denotes the degree of a vertex \( u \) in \( G \) and \( E(G) \) represents the edge set of \( G \). Let \( G^* \) be a graph having the maximum value of \( cM_2 \) among all connected graphs of order \( n \). Furtula and Oz [MATCH Commun. Math. Comput. Chem. 93 (2025) 247--263] conjectured that \( G^* \) is the join \( K_k+\overline{K}_{n-k} \) of the complete graph \( K_k \) of order \( k \) and the complement \( \overline{K}_{n-k} \) of the complete graph \( K_{n-k} \) such that the inequality \( k<\lceil n/2 \rceil \) holds. We prove that (i) the maximum degree of \( G^* \) is \( n-1 \) and (ii) no two vertices of minimum degree in \( G^* \) are adjacent; both of these results support the aforementioned conjecture. We also prove that the number of vertices of maximum degree in \( G^* \), say \( k \), is at most \( -\frac{2}{3}n+\frac{3}{2}+\frac{1}{6}\sqrt{52n^2-132n+81} \), which implies that \( k < 5352n / 10000 \). Furthermore, we establish results that support the conjecture under consideration for certain bidegreed and tridegreed graphs. In the aforesaid paper, it was also mentioned that determining the \( k \) as a function of the \( n \) is far from being an easy task; we obtain the values of \( k \) for \( 5\le n\le 149 \) in the case of certain bidegreed graphs by using computer software and found that the resulting sequence of the values of \( k \) does not exist in "The On-Line Encyclopedia of Integer Sequences" (an online database of integer sequences).