  Self-testing is a powerful method for certifying quantum systems. Initially
proposed in the device-independent (DI) setting, self-testing has since been
relaxed to the semi-device-independent (semi-DI) setting. In this study, we
focus on the self-testing of a specific type of non-projective qubit
measurements belonging to a one-parameter family, using the semi-DI
prepare-and-measure (PM) scenario. Remarkably, we identify the simplest PM
scenario discovered so far, involving only four preparations and four
measurements, for self-testing the fourth measurement. This particular
measurement is a four-outcome non-projective positive operator-valued measure
(POVM) and falls in the class of semisymmetric informationally complete
(semi-SIC) POVMs introduced by Geng et al. [Phys. Rev. Lett. 126, 100401
(2021)]. To achieve this, we develop analytical techniques for semi-DI
self-testing in the PM scenario. Our results shall pave the way towards
self-testing any extremal qubit POVM within a potentially minimal PM scenario.
