Tilted-unduloid conjecture for the spherical-helicoid deformation

Let $H>\frac{1}{2}$, let $c\in(0,1]$, and let $f^c$ be the spherical helicoid under consideration. Define

$$h_1:=f^c(\cdot,0),\qquad h_2^\alpha:=f^c(\cdot,\alpha),$$

for $\alpha\in I$, and let $\mathcal{M}(h_1,h_2^\alpha)$ denote the relevant set of embedded minimal annuli bounded by these curves. Let $\alpha_0$ be the exceptional parameter, and let $\widetilde{\Sigma^\alpha}$ denote the sister surface in $\mathbb{H}^2\times\mathbb{R}$.

Tilted-unduloid conjecture. For every $\alpha\in I\setminus\{\alpha_0\}$, there exists $\Sigma^\alpha\in\mathcal{M}(h_1,h_2^\alpha)$ such that $\widetilde{\Sigma^\alpha}$ extends to a tilted unduloid in $\mathbb{H}^2\times\mathbb{R}$.

The conjecture extends the perturbative existence result proved in the paper, which establishes the conclusion only for parameters sufficiently close to the exceptional value. The source gives no resolution for all $\alpha\in I\setminus\{\alpha_0\}$ together.