Let \(G\) be a simple graph with vertex set \(V(G) = \{v_1,\,v_2, \ldots,\,v_n\}\) and edge set \(E(G)\), where \(|V(G)|=n\) and \(|E(G)|=m\). Molecular descriptors play a significant role in quantitative studies of structure-property and structure-activity relationships. One of the popular degree-based topological indices, the Sombor index \((SO)\), is a chemically useful descriptor. The Sombor index of a graph \(G\) is defined as \[ SO(G)=\sum\limits_{v_iv_j\in E(G)}\,\sqrt{d_i^2+d_j^2}, \] where \(d_i\) is the degree of the vertex \(v_i\in V(G)\). The Sombor matrix of \(G\), denoted by \(SM(G)\), is defined as the \(n \times n\) matrix whose \((i,j)\)-entry is \(\sqrt{d_i^2 + d_j^2}\) if \(v_iv_j\in E(G)\), and \(0\) otherwise. Let the eigenvalues of the Sombor matrix \(SM(G)\) be \(\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_n\). Very recently, Rabizadeh, Habibi and Gutman {[Some notes on Sombor index of graphs, MATCH Commun. Math. Comput. Chem. 93 (2025) 853-859]} proposed a conjecture about the Sombor index of graphs, stated as follows:

\((a)\) \[ m|\sigma_n| = -m\sigma_n \geq SO(G). \] \((b)\) If \(G\) is connected, then the equality in \((a)\) holds if and only if \(G\) is a complete graph. In the general case, equality holds if and only if \(G\) consists of mutually isomorphic complete graphs and some (or no) isolated vertices. In this paper, we provide a complete solution to the above conjecture.