Let $\mathcal X$ be the space of type-preserving $\mathrm{SL}(2,\mathbb{C})$
characters of the punctured torus $T$. The Bowditch space
$\mathcal{X}_{\mathrm{BQ}}$ is the largest open subset of
$\mathcal{X}$ on which the mapping class group acts properly
discontinuously, this is characterized by two simple conditions
called the BQ-conditions. In this note, we show that $[\rho]
\in \operatorname{int}(\mathcal{X} \setminus \mathcal{X}_{\mathrm{BQ}})$
if there exists an essential simple closed curve $X$ on $T$
such that $|\tr\rho(X)|<0.5$.