  We investigate the solution space of the $\beta$-deformed Quantum Spectral
Curve by studying a sample of solutions corresponding to single-trace operators
that in the undeformed theory belong to the Konishi multiplet. We discuss how
to set the precise boundary conditions for the leading Q-system for a given
state, how to solve it, and how to build perturbative corrections to the
$\mathbf{P}\mu$-system. We confirm and add several loop orders to known results
in the literature.
