The frustrated one-dimensional (1D) quantum magnet LiCuSbO$_4$ is one rare realization of the $J_1-J_2$ spin chain model with an easily accessible saturation field, formerly estimated to 12~T. Exotic multipolar nematic phases were theoretically predicted in such compounds just below the saturation field, but without unambiguous experimental observation so far. In this paper we present extensive experimental research of the compound in the wide temperature (30mK$-$300K) and field (0$-$13.3T) range by muon spin rotation ($mu$SR), $^7$Li nuclear magnetic resonance (NMR) and magnetic susceptibility (SQUID). $mu$SR experiments in zero magnetic field demonstrate the absence of long range 3D ordering down to 30mK. Together with former heat capacity data [S.E. Dutton emph{et al}, Phys. Rev. Lett. 108, 187206 (2012)], magnetic susceptibility measurements suggest short range correlated vector chiral phase in the field range $0-4$T. In the intermediate field values (5$-$12T), the system enters in a 3D ordered spin density wave phase with 0.75$mu_B$ per copper site at lowest temperatures (125mK), estimated by NMR. At still higher field, the magnetization is found to be saturated above 13T where the spin lattice $T_1^{-1}$ relaxation reveals a spin gap estimated to 3.2(2)K. We narrow down the possibility of observing a multipolar nematic phase to the range 12.5$-$13T.
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