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We study immersed tori in $3$-space minimizing the Willmore energy in theirrespective conformal class. Within the rectangular conformal classes$\;(0,b)\;$ with $\;b \sim 1\;$ the homogenous tori $\;f^b\;$ are known to bethe unique constrained Willmore minimizers (up to invariance). In this paper wegeneralize this result and show that the candidates constructed in\cite{HelNdi2} are indeed constrained Willmore minimizers in certainnon-rectangular conformal classes $\;(a,b).\;$ Difficulties arise from the factthat these minimizers are non-degenerate for $\;a \neq 0\;$ but smoothlyconverge to the degenerate homogenous tori $\;f^b\;$ as $\;a \longrightarrow0.\;$ 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 $\;b \sim 1,\;$ $b \neq 1.$

First explicit constrained Willmore minimizers of non-rectangular conformal class