  In this article, we analyze the stability of the parallel surface problem for
semilinear equations driven by the fractional Laplacian. We prove a
quantitative stability result that goes beyond that previously obtained in
[Cir+23].
  Moreover, we discuss in detail several techniques and challenges in obtaining
the optimal exponent in this stability result. In particular, this includes an
upper bound on the exponent via an explicit computation involving a family of
ellipsoids. We also sharply investigate a technique that was proposed in
[Cir+18] to obtain the optimal stability exponent in the quantitative estimate
for the nonlocal Alexandrov's soap bubble theorem, obtaining accurate estimates
to be compared with a new, explicit example.
