The access control problem in a hierarchy can be solved by using a hierarchical key assignment scheme, where each class is assigned an encryption key and some private information. A formal security analysis for hierarchical key assignment schemes has been traditionally considered in two different settings, i.e., the unconditionally secure and the computationally secure setting, and with respect to two different notions: security against key recovery (KR-security) and security with respect to key indistinguishability (KI-security), with the latter notion being cryptographically stronger. Recently, Freire, Paterson and Poettering proposed strong key indistinguishability (SKI-security) as a new security notion in the computationally secure setting, arguing that SKI-security is strictly stronger than KI-security in such a setting. In this paper we consider the unconditionally secure setting for hierarchical key assignment schemes. In such a setting the security of the schemes is not based on specific unproven computational assumptions, i.e., it relies on the theoretical impossibility of breaking them, despite the computational power of an adversary coalition. We prove that, in this setting, SKI-security is not stronger than KI-security, i.e., the two notions are fully equivalent from an information-theoretic point of view.