Enhancement of spin-lattice relaxation rate of $\beta$$^{\prime\prime}$-(BEDT-TTF)$_{2}$SF$_{5}$CH$_{2}$CF$_{2}$SO$_{3}$ in its high-field superconducting phase

We report a $^{13}$C NMR study of an organic charge transfer salt $\beta$$^{\prime\prime}$-(BEDT-TTF)$_{2}$SF$_{5}$CH$_{2}$CF$_{2}$SO$_{3}$, focusing on the spin dynamics in its high-field inhomogeneous superconducting phase, a putative FFLO state. A noticeable enhancement of the spin-lattice relaxation rate beyond the normal state value is observed within the limits of the FFLO phase for a range of fields at T=750mK, resembling similar features previously observed in the related compound $\kappa$-ET. This enhancement is suppressed upon cooling, and its freeze-out is investigated in detail. Moreover, the angular dependence of the relaxation rate in the vicinity of the aligned in-plane field condition is probed, in an effort to inform the origin of the observed behavior with orbital or hyperfine effects.