<p>In their seminal work on truncated sums of theta functions, Andrews and Merca established the truncated sums of Euler’s pentagonal number theorem and proved an infinite families of linear inequalities for ordinary partition function. Motivated by their work, linear inequalities involving various partition functions have been established in recent years. Recently, Merca investigated two partition functions whose generating functions are the Göllnitz–Gordon functions. He also posed a conjecture on linear inequalities for the coefficients of the Göllnitz–Gordon functions. In this paper, we confirm Merca’s conjecture based on a classical result owing to Pólya and Szegö.</p>