Singular apparent-space conjecture for even configurations on elliptic curves

Let $E_\tau$ be the elliptic curve with parameter $\tau\in\mathbb{H}$, and let $\Sigma=\{p_1,\ldots,p_n\}\subseteq E_\tau$, where $n$ is even. Let $AP_\Sigma(\tau)$ be the associated apparent space and $\mathcal{SM}_\Sigma(\tau)$ the corresponding solution space. A point $(\mathbb{T},B)$ is called singular when it is a singular point of $AP_\Sigma(\tau)$. Singular apparent-space conjecture. If $AP_\Sigma(\tau)$ includes a singular point $(\mathbb{T},B)$, then $\mathcal{SM}_\Sigma(\tau)$ contains a subset isomorphic to

$$SL(2,\mathbb{C})/SU(2).$$

The claim extends the observed half-period phenomenon from four singularities to every even number of marked points, but no resolution is supplied in the source.