Evenness conjecture for planar-ended minimal surfaces on the sphere

Let $S^2$ be the Riemann sphere, let $p_1,\ldots,p_d$ be distinct points, and let

$$\vec{\Phi}:S^2\setminus\{p_1,\ldots,p_d\}\longrightarrow \mathbb{R}^3$$

be a non-planar minimal surface with embedded planar ends. Evenness conjecture. Then $d\geq 4$ is even.

This concerns the possible number of embedded planar ends of complete minimal surfaces of genus zero in $\mathbb{R}^3$. The surrounding results establish nonexistence for nine embedded planar ends and suggest that the same restriction may hold for every odd number at least $11$, but the stated claim is not accompanied by a resolution here.