  Ejiri's torus in $S^5$ is the first example of Willmore surface which is not
conformally equivalent to any minimal surface in any space forms. Li and
Vrancken classified all Willmore surfaces of tensor product in $S^{n}$ by
reducing them into elastic curves in $S^3$, and the Ejiri torus appeared as a
special example. In this paper, we first prove that among all Willmore tori of
tensor product, the Willmore functional of the Ejiri torus in $S^5$ attains the
minimum $2\pi^2\sqrt{3}$. Then we show that all Willmore tori of tensor product
are unstable when the co-dimension is big enough. We also show that the Ejiri
torus is unstable in $S^5$. Moreover, similar to Li and Vrancken, we classify
all constrained Willmore surfaces of tensor product by reducing them with
elastic curves in $S^3$. All constrained Willmore tori obtained this way are
also shown to be unstable when the co-dimension is big enough.
  We conjecture that a Willmore torus having Willmore functional between
$2\pi^2$ and $2\pi^2\sqrt{3}$ is either the Clifford torus, or the Ejiri torus.
