Summary: Recently in [D. Sain and K. Paul, Linear Algebra Appl. 439, No. 8, 2448–2452 (2013; Zbl 1291.46024)] it was proved that if \(T\) is a linear operator on a finite dimensional real normed linear space \(\mathbb{X}\) such that \(T\) attains norm only on \(\pm D\), where \(D\) is a connected closed subset of \(S_{\mathbb{X}}\), then \(T\) satisfies the Bhatia-Šemrl (BŠ) property [R. Bhatia and P. Šemrl, Linear Algebra Appl. 287, No. 1–3, 77–85 (1999; Zbl 0937.15023)], i.e., for \(A \in \mathbb{L}(\mathbb{X})\), \(T \bot_B A\) implies that there exists \(x \in D\) such that \(T x \bot_B A x\). Here we explore the converse of the above result. We prove that in a real normed linear space \(\mathbb{X}\) of dimension \(2\), a linear operator \(T\) satisfies the BŠ property if and only if the set of unit vectors on which \(T\) attains norm is connected in the corresponding projective space \(\mathbb{R} P^1 \equiv S_{\mathbb{X}} / \{x \sim - x \}\). Motivated by the result in 2 dimensions, we conjecture that this characterization of BŠ property is true in general \(n\)-dimensional real normed linear spaces. We further prove that if the space \(\mathbb{X}\) is strictly convex, then the set of operators in \(\mathbb{L}(\mathbb{X})\) which satisfy the BŠ property is dense in \(\mathbb{L}(\mathbb{X})\).