Density of the rational obstructed boundary in the obstructed boundary

Let $\mathcal{H}$ be a hyperbolic component of disjoint type. Define its obstructed boundary by

$$\partial^\infty\mathcal{H}=\{\rho\in\partial\mathbb{D}^{2d-2}:\text{ there exists }[f_n]\in\mathcal{H}\text{ with }[f_n]\to\infty\text{ in }\operatorname{\mathcal{M}}_{d,\text{fm}}\text{ and }\boldsymbol{\rho}([f_n])\to\rho\}.$$

Let $\partial^\infty_{\mathbb{Q}}\mathcal{H}$ be the subset of $\partial^\infty\mathcal{H}$ for which all indifferent multipliers are rational, of the form $e^{2\pi i p/q}$. The rational-boundary density conjecture.

$$\partial^\infty\mathcal{H}=\overline{\partial^\infty_{\mathbb{Q}}\mathcal{H}}.$$

Consequently, the natural extension of the multiplier map identifies the regular boundary as

$$\partial_{\operatorname{reg}}\mathcal{H}\cong\partial\mathbb{D}^{2d-2}-\overline{\partial^\infty_{\mathbb{Q}}\mathcal{H}}-\boldsymbol{\rho}(\partial_{\operatorname{exc}}\mathcal{H}).$$

This would describe the boundary at infinity through obstructed geometrically finite maps and support an extension of the quadratic-like theory using sector renormalization, but the statement is presented as a conjecture.