Concealing the identity through speaker anonymization is essential in various situations. This study focuses on investigating how stuttering affects the anonymization process. Two scenarios are considered: preserving the pathology in the diagnostic/remote treatment context and obfuscating the pathology. The paper examines the effectiveness of three state-of-the-art approaches in achieving high anonymization, as well as the preservation of dysfluencies. The findings indicate that while a speaker conversion method may not achieve perfect anonymization (Baseline 27.25% EER and F0 Delta 32.63% EER), it does preserve the pathology. This effect was objectively evaluated by performing a stuttering classification. Although this solution may be useful in a remote treatment scenario for speech pathologies, it presents a vulnerability in anonymization. To address this issue, we propose an alternative approach that uses automatic speech recognition and text-based speech synthesis to avoid re-identification (48.27% EER).