Can the use of the Leggett-Garg inequality enhance security of the BB84 protocol?

Prima facie, there are good reasons to answer in the negative the question posed in the title: the Bennett-Brassard 1984 (BB84) protocol is provably secure subject to the assumption of trusted devices, while the Leggett-Garg-type inequality (LGI) does not seem to be readily adaptable to the device independent (DI) or semi-DI scenario. Nevertheless, interestingly, here we identify a specific device attack, which has been shown to render the standard BB84 protocol completely insecure, but against which our formulated LGI-assisted BB84 protocol (based on an appropriate form of LGI) is secure.