  We continue to develop Bootstrability -- a method merging Integrability and
Conformal Bootstrap to extract CFT data in integrable conformal gauge theories
such as $\mathcal{N}$=4 SYM. In this paper, we consider the 1D defect CFT
defined on a $\frac{1}{2}$-BPS Wilson line in the theory, whose
non-perturbative spectrum is governed by the Quantum Spectral Curve (QSC). In
addition, we use that the deformed setup of a cusped Wilson line is also
controlled by the QSC. In terms of the defect CFT, this translates into two
nontrivial relations connecting integrated 4-point correlators to cusp spectral
data, such as the Bremsstrahlung and Curvature functions -- known analytically
from the QSC. Combining these new constraints and the spectrum of the $10$
lowest-lying states with the Numerical Conformal Bootstrap, we obtain very
sharp rigorous numerical bounds for the structure constant of the first
non-protected state, giving this observable with seven digits precision for the
't Hooft coupling in the intermediate coupling region
$\frac{\sqrt{\lambda}}{4\pi}\sim 1$, with the error decreasing quickly at large
't Hooft coupling. Furthermore, for the same structure constant we obtain a
$4$-loop analytic result at weak coupling. We also present results for excited
states.
