By [W. Parry (1964)], every mixing subshift of finite type has a unique measure of maximal entropy (mme). A natural question is, how close an invariant measure must be to the mme when its metric entropy is close to the topological entropy? [Shirali Kadyrov (2015)] bound the distance between an invariant measure \(\mu\) and the mme in terms of its associated entropy \(h_{\sigma}(\mu)\) and \(h_{top}\) in a weak* sense. We will describe a strategy for extending this result to Ornstein's \(d\)-bar metric. This is a joint work with Dr. Vaughn Climenhaga.