A strong designated verifier signature (SDVS) is a variation of traditional digital signatures, since it allows a signer to designate an intended receiver as the verifier rather than anyone. To do this, a signer must incorporate the verifier's public key with the signing procedure such that only the intended receiver could verify this signature with his/her private key. Such a signature further enables a designated verifier to simulate a computationally indistinguishable transcript intended for himself. Consequently, no one can identify the real signer's identity from a candidate signer and a designated verifier, which is referred to as the property of signer ambiguity. A strong notion of signer ambiguity states that no polynomial-time adversary can distinguish the real signer of a given SDVS that is not received by the designated verifier, even if the adversary has obtained the signer's private key. In 2013, Islam and Biswas proposed a provably secure certificateless strong designated verifier signature (CL-SDVS) scheme based on bilinear pairings. In this paper, we will demonstrate that their scheme fails to satisfy strong signer ambiguity and must assume a trusted private key generator (PKG). In other words, their CL-SDVS scheme is vulnerable to both key-compromise and malicious PKG attacks. Additionally, we present an improved variant to eliminate these weaknesses.