Both the long-duration gamma-ray bursts (LGRBs) and the Type I superluminous supernovae (SLSNe~I) have been proposed to be primarily powered by central magnetars. A correlation, proposed between the initial spin period ($P_0$) and the surface magnetic field ($B$) of the magnetars powering the X-ray plateaus in LGRB afterglows, indicates a possibility that the magnetars have reached an equilibrium spin period due to the fallback accretion. The corresponding accretion rates are inferred as $dot{M}approx10^{-4}-10^{-1}$ M$_odot$ s$^{-1}$, and this result holds for the cases of both isotropic and collimated magnetar wind. For the SLSNe~I and a fraction of engine-powered normal type Ic supernovae (SNe~Ic) and broad-lined subclass (SNe~Ic-BL), the magnetars could also reach an accretion-induced spin equilibrium, but the corresponding $B-P_0$ distribution suggests a different accretion rate range, i.e., $dot{M}approx 10^{-7}-10^{-3}$ M$_odot$ s$^{-1}$. Considering the effect of fallback accretion, magnetars with relatively weak fields are responsible for the SLSNe~I, while those with stronger magnetic fields could lead to SNe~Ic/Ic-BL. Some SLSNe~I in our sample could arise from compact progenitor stars, while others that require longer-term accretion may originate from the progenitor stars with more extended envelopes or circumstellar medium.
Download