  We analyze the linear stability of the base state of the problem of coupled
flow and deformation in a long and shallow rectangular soft hydraulic conduit
with a thick top wall. Specifically, the steady base state is computed at low
but finite Reynolds number. Then, we show that with the upstream flux fixed and
the outlet pressure set to gauge, the flow is linearly stable to infinitesimal
flow-wise perturbations. Multiple oscillatory but stable eigenmodes are
computed in a range of the reduced Reynolds number, $\hat{Re}$, and the
so-called fluid--structure interaction (FSI) parameter, $\lambda$, indicating
the stiffness of this FSI system. These results provide a framework to address,
in future work, the individual effects of various aspects of two-way FSI
coupling on instability and flow transition in soft hydraulic conduits.
