Sharding has emerged as one of the common techniques to address the scalability problems of blockchain systems. To this end, various sharding techniques for blockchain systems have been proposed in the literature. When sharded blockchains process personal data, the data controllers and the data processors associated with the sharded blockchains need to be compliant with the General Data Protection Regulation (GDPR). To this end, this article makes the first attempt to address the following key question: to what extent the existing techniques developed by different communities such as the distributed computing community, the distributed systems community, the database community, identity and access control community and the dependability community can be used by the data controllers and data processors for complying with the GDPR requirements of data subject rights in sharded blockchains? As part of answering this question, this article argues that there is a need for cross-disciplinary research towards finding optimal solutions for implementing the data subject rights in sharded blockchains.