  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.
