Conjectural description of the smoothing-component moduli space

Assume that $D$ is negative definite. Let $D_1,\ldots,D_k$ be the irreducible components of $D$, and define

$$Q=\langle D_1,\ldots,D_k\rangle^\perp\subset\operatorname{Pic}Y.$$

Let $\mathcal{C}$ be the interior of $\overline{K(Y_{\mathrm{gen}})}\cap Q$ inside $\operatorname{Pic}Y\otimes\mathbb{R}=H^2_{dR}(Y;\mathbb{R})$, where $K(Y_{\mathrm{gen}})$ is the generic Kähler cone; let $\Phi$ be the set of all roots in $Q$; and let $\mathrm{Adm}$ be the subgroup of admissible lattice automorphisms of $\operatorname{Pic}Y$ preserving the classes $[D_i]$ and $\mathcal{C}$. Smoothing-moduli conjecture. There is a natural isomorphism

$$\mathbf{S}\simeq\Big((Q_{\mathbb{R}}+i\mathcal{C})\backslash\bigcup_{\alpha\in\Phi,\ k\in\mathbb{Z}}\{x\mid\langle x,\alpha\rangle=k\}\Big)/(Q\rtimes\mathrm{Adm}).$$

For $k\leq5$, the source says this statement is a theorem of Looijenga; the conjectural status concerns the general cases under discussion.