  We reconsider $D\geq4$ dimensional asymptotically flat eternal Schwarzschild
black hole, and focus on the situation where the inner boundary of the
radiation region is chosen to be near the horizon (i.e. $\beta \ll1$). The
tension between the near-horizon condition and the short-distance approximation
emerges in large dimensions in $[JHEP 06 (2020) 085]$. We remove this tension
by introducing a more proper near horizon condition, thus the resulting island
solution is well-behaved in any $D\geq4$ dimensional spacetime. Interestingly,
a novel constraint is obtained in this situation as required by the existence
of the island solution, which directly leads to the constraints on the size of
the Schwarzschild black hole, the position of the inner boundary for the
radiation region, or the value of $c\cdot\tilde{G}_{N}$ in any $D\geq4$
dimension. When considering the large $D$ limit, the constraint on the size of
the Schwarzschild black hole obtained in this situation is in agreement with
the result given in $[Phys.Rev.D 102 (2020) 2, 026016]$. We interpret these as
the unitary constraints implied by the presence of island in semiclassical
gravity.
