For large initial rectilinear displacements, we investigate free response of a linear oscillator inertially coupled to a rotational nonlinear energy sink (NES). Absent direct rectilinear damping, two ranges of orbitally stable solutions are accessible, in contrast to one range at lower, intermediate energies (2021, J. Appl. Mech. 88, 121009). As at lower initial energies, fractal and riddled basins of attraction are identified for both previously considered numerical combinations of the two dimensionless parameters (characterizing rotational damping, and coupling of rotational to rectilinear motion). In this case, however, the final amplitude, in addition to the final angular orientation, displays initial-condition sensitivity. For one parameter combination, most motionless initial conditions (MICs, with the masses of the linear oscillator and NES both initially at rest) lead to a discrete set of “special” orbitally stable solutions with one of two amplitudes, and no MICs lead to complete dissipation of initial energy. For the other combination, the results are similar to those at intermediate initial energies, with no MICs leading to special solutions, and many leading to complete dissipation. Results for both parameter combinations strongly suggest that no qualitatively new behavior occurs for initial energies beyond those considered here, with nonrotating asymptotic special-amplitude solutions having larger rectilinear amplitude either not existing or existing in only very small parts of the MIC space. With direct damping of rectilinear motion, transient chaos leads to initial-condition sensitivity of settling time, with the case of no direct rectilinear damping providing good guidance to damped-case behavior.