  We study immersed tori in $3$-space minimizing the Willmore energy in their
respective conformal class. Within the rectangular conformal classes
$\;(0,b)\;$ with $\;b \sim 1\;$ the homogenous tori $\;f^b\;$ are known to be
the unique constrained Willmore minimizers (up to invariance). In this paper we
generalize this result and show that the candidates constructed in
\cite{HelNdi2} are indeed constrained Willmore minimizers in certain
non-rectangular conformal classes $\;(a,b).\;$ Difficulties arise from the fact
that these minimizers are non-degenerate for $\;a \neq 0\;$ but smoothly
converge to the degenerate homogenous tori $\;f^b\;$ as $\;a \longrightarrow
0.\;$ As a byproduct of our arguments, we show that the minimal Willmore energy
$\;\omega(a,b)\;$ is real analytic and concave in $\;a \in (0, a^b)\;$ for some
$\;a^b>0\;$ and fixed $\;b \sim 1,\;$ $b \neq 1.$
