For a graph \( G \) with edge set \( E \), let \( d(u) \) denote the degree of a vertex \( u \) in \( G \). The diminished Sombor (DSO) index of \( G \) is defined as \( DSO(G)=\sum_{uv\in E}\sqrt{(d(u))^2+(d(v))^2}(d(u)+d(v))^{-1} \). The cyclomatic number of a graph is the smallest number of edges whose removal makes the graph acyclic. A connected graph of maximum degree at most \( 4 \) is known as a molecular graph. The primary motivation of the present study comes from a conjecture, concerning the minimum DSO index of fixed-order connected graphs with cyclomatic number \( 3 \), posed in the recent paper [F. Movahedi, I. Gutman, I. Red\v{z}epovi\'c, B. Furtula, Diminished Sombor index, {\it MATCH Commun. Comput. Chem.\/} {\bf 95} (2026) 141--162]. The present paper gives all graphs minimizing the DSO index among all molecular graphs of order \( n \) with cyclomatic number \( \ell \), provided that \( n\ge 2(\ell-1)\ge4 \).