Spectrum-aware debiasing conjecture for non-separable convex penalties

Let $\vec{h}:\mathbb{R}^p\mapsto\mathbb{R}$ be a proper, closed, twice-differentiable convex penalty, or admit a twice-differentiable extension, and let

$$\hat{\bm{\beta}} \in \underset{\mathbf{b} \in \mathbb{R}^p}{\arg\min}\; \frac{1}{2}\|\mathbf{y}-\mathbf{X}\mathbf{b}\|^2+\vec{h}(\mathbf{b}).$$

Define $\hat{\bm{\beta}}^u$, $\widehat{\mathsf{adj}}$, and $\hat{\tau}_*$ by the formulas preceding the conjecture, with $\widehat{\mathsf{adj}}$ satisfying

$$\frac{1}{p}\sum_{i=1}^p\frac{1}{\frac{d_i^2-\widehat{\mathsf{adj}}}{p}\operatorname{Tr}\left(\left(\widehat{\mathsf{adj}}\cdot\mathbf{I}_p+\bm{\Sigma}^{-1}\left(\nabla^2\vec{h}(\hat{\bm{\beta}})\right)\right)^{-1}\right)+1}=1.$$

Spectrum-aware debiasing conjecture. Under suitable conditions, there is a unique solution $\widehat{\mathsf{adj}}$ of this equation and

$$\hat{\tau}_*^{-1/2}(\hat{\bm{\beta}}^u-\bm{\beta}^\star)=\bm{\Sigma}^{1/2}\mathbf{z}+O\left(p^{-1/2}\right),$$

where $\mathbf{z}\sim N(\mathbf{0},\mathbf{I}_p)$ and $O\left(p^{-1/2}\right)$ denotes a vector $\mathbf{v}\in\mathbb{R}^p$ satisfying $\frac{1}{p}\|\mathbf{v}\|^2\to0$ almost surely as $p\to\infty$. This conjecture proposes an asymptotically Gaussian distribution for the debiased estimator under non-separable penalties; the uniqueness and the stated approximation remain open.