  For two inner functions $\vartheta,\varphi\in H^\infty$, we give a simple
sufficient condition for the system $\vartheta^m,\; \varphi^n$,
$m,n\in\mathbb{Z}$, to be complete in the weak-$^*$ topology of
$L^\infty(\mathbb{T})$. To be precise, we show that this system is complete
whenever there is an arc $I$ of the unit circle $\mathbb{T}$ such that
$\vartheta$ is univalent on $I$ and $\varphi$ is univalent on
$\mathbb{T}\setminus I$. As an application of this result, we describe a class
of analytic curves $\Gamma$ such that $(\Gamma, \mathcal{X})$ is a Heisenberg
uniqueness pair, where $\mathcal{X}$ is the lattice cross
$\{(m,n)\in\mathbb{Z}^2:\, mn=0\}$. Our main result extends a theorem of
Hedenmalm and Montes-Rodr\'iguez for atomic inner functions with one
singularity.
