Digital twin technology has become of noticeable interest in the automotive industry. There is a growing need for smarter services that utilize the unique capabilities of digital twins, ranging from computer-aided remote control to cloud-based fleet coordination. The safety-critical nature of automotive systems imposes high quality requirements on software technology, including digital twins. Meeting such requirements starts with a proper software architecture that underpins automotive services. However, the relative scarcity of digital twin architectural guidelines poses a severe challenge for engineering automotive digital twins. At the time of writing this paper, the only available digital twin architectural standard is the one defined in ISO 23247, which itself has not been developed specifically for automotive systems and has demonstrated limitations when applied outside of its original manufacturing scope. Still, this reference architecture might be one of the few feasible starting points for automotive digital twins. In this work, we investigate the fitness for purpose of the ISO 23247 reference architecture in developing automotive digital twins. Through the case study of a 1/10th-scale autonomous vehicle, we identify the strengths and limitations of the reference architecture and distill key future directions for researchers, practitioners, and standard developers.