  We study the scaling limit of a statistical system, which is a special case
of the integrable inhomogeneous six-vertex model. It possesses
$U_q\big(\mathfrak{sl}(2)\big)$ invariance due to the choice of open boundary
conditions imposed. An interesting feature of the lattice theory is that the
spectrum of scaling dimensions contains a continuous component. By applying the
ODE/IQFT correspondence and the method of the Baxter $Q$ operator the
corresponding density of states is obtained. In addition, the partition
function appearing in the scaling limit of the lattice model is computed, which
may be of interest for the study of nonrational CFTs in the presence of
boundaries. As a side result of the research, a simple formula for the matrix
elements of the $Q$ operator for the general, integrable, inhomogeneous
six-vertex model was discovered, that has not yet appeared in the literature.
It is valid for a certain one parameter family of diagonal open boundary
conditions in the sector with the $z\,$-projection of the total spin operator
being equal to zero.
