Singular apparent-space conjecture for half-period configurations

Let $\tau\in\mathbb{H}$, let $\Sigma=E_\tau[2]$, and let $(\alpha_1,\ldots,\alpha_4)\in(2\mathbb{Z})^4$. The apparent space $AP_\Sigma(\tau)$ parametrizes the relevant apparent data, and $\mathcal{SM}_\Sigma(\tau)$ denotes 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

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

and the number of such copies is exactly the number of singular points of $AP_\Sigma(\tau)$. This predicts that each singular apparent datum contributes one such family of solutions; the paper notes that in the half-period case the singular fiber is of this form, while the general assertion is conjectural.