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Location obfuscation functions generated by existing systems for ensuring location privacy are monolithic and do not allow users to customize their obfuscation range. This can lead to the user being mapped in undesirable locations (e.g., shady neighborhoods) to the location-requesting services. Modifying the obfuscation function generated by a centralized server on the user side can result in poor privacy as the original function is not robust against such updates. Users themselves might find it challenging to understand the parameters involved in obfuscation mechanisms (e.g., obfuscation range and granularity of location representation) and therefore struggle to set realistic trade-offs between privacy, utility, and customization. In this paper, we propose a new framework called, CORGI, i.e., CustOmizable Robust Geo-Indistinguishability, which generates location obfuscation functions that are robust against user customization while providing strong privacy guarantees based on the Geo-Indistinguishability paradigm. CORGI utilizes a tree representation of a given region to assist users in specifying their privacy and customization requirements. The server side of CORGI takes these requirements as inputs and generates an obfuscation function that satisfies Geo-Indistinguishability requirements and is robust against customization on the user side. The obfuscation function is returned to the user who can then choose to update the obfuscation function (e.g., obfuscation range, granularity of location representation). The experimental results on a real dataset demonstrate that CORGI can efficiently generate obfuscation matrices that are more robust to the customization by users.