The regularity conjecture for the Real global group law

Let $\mathbf{R}$ denote the Real global group law constructed from the Real globally oriented spectrum $\mathbf{MR}$. For each torus $\mathbb{T}^n$, let $\Phi^{\mathbb{T}^n}\mathbf{R}$ denote its geometric-fixed-point target, and let $\mathbf{R}(\mathbb{T}^n)$ be the corresponding value of the global group law. Regularity Conjecture. The global group law $\mathbf{R}$ is regular in the sense of Hausmann's Definition 5.9. In particular, the maps

$$\mathbf{R}(\mathbb{T}^n)\longrightarrow\Phi^{\mathbb{T}^n}\mathbf{R}$$

are injective. Regularity is proposed as an alternative conjectural assumption for proving injectivity of the restriction maps for augmented tori. The source does not establish it, so its status remains open.