A study of $C\!P$ violation in the decays $B^\pm\to[K^+K^-\pi^+\pi^-]_D h^{\pm}$ ($h = K, \pi$) and $B^\pm\to[\pi^+\pi^-\pi^+\pi^-]_D h^{\pm}$

The first study of $C\!P$ violation in the decay mode $B^\pm\to[K^+K^-\pi^+\pi^-]_D h^\pm$, with $h=K,\pi$, is presented, exploiting a data sample of proton-proton collisions collected by the LHCb experiment that corresponds to an integrated luminosity of $9$ fb$^{-1}$. The analysis is performed in bins of phase space, which are optimised for sensitivity to local $C\!P$ asymmetries. $C\!P$-violating observables that are sensitive to the angle $\gamma$ of the Unitarity Triangle are determined. The analysis requires external information on charm-decay parameters, which are currently taken from an amplitude analysis of LHCb data, but can be updated in the future when direct measurements become available. Measurements are also performed of phase-space integrated observables for $B^\pm\to[K^+K^-\pi^+\pi^-]_D h^\pm$ and $B^\pm\to[\pi^+\pi^-\pi^+\pi^-]_D h^\pm$ decays.