  This note is concerned with the Bianchi-Egnell inequality, which quantifies
the stability of the Sobolev inequality, and its generalization to fractional
exponents $s \in (0, \frac{d}{2})$. We prove that in dimension $d \geq 2$ the
best constant \[ c_{BE}(s) = \inf_{f \in \dot{H}^s(\mathbb R^d) \setminus
\mathcal M} \frac{\|(-\Delta)^{s/2} f\|_{L^2(\mathbb R^d)}^2 - S_{d,s}
\|f\|_{L^{2^*}(\mathbb R^d)}^2}{\text{dist}_{\dot{H}^s(\mathbb R^d)}(f,
\mathcal M)^2} \] is strictly smaller than the spectral gap constant
$\frac{4s}{d+2s+2}$ associated to sequences which converge to the manifold
$\mathcal M$ of Sobolev optimizers. In particular, $c_{BE}(s)$ cannot be
asymptotically attained by such sequences. Our proof relies on a precise
expansion of the Bianchi-Egnell quotient along a well-chosen sequence of test
functions converging to $\mathcal M$.
