Conformal limits in Cayley components and $\Theta$-positive opers

We study Gaiotto's conformal limit for the $G^{\mathbb{R}}$-Hitchin equations, when $G^{\mathbb{R}}$ is a simple real Lie group admitting a $\Theta$-positive structure. We identify a family of flat connections coming from certain solutions to the equations for which the conformal limit exists and admits the structure of an oper. We call this new class of opers appearing in the conformal limit $\Theta$-positive opers. The two families involved are parameterized by the same base space. This space is a generalization of the base of Hitchin's integrable system in the case when the structure group is a split real group.