Limiting FDP and TPP conjecture for TPoP and CPoP

Let Assumption hold. In the scalar model, let $\Phi={\mathcal P}(Y;\Pi,\tau_\star)$ be the local false-discovery rate, where $(\beta_0,\Phi)$ has the joint distribution induced by the scalar model. For a threshold $t\in[0,1]$ and tuning parameter $\lambda$, let $t_\star(\lambda)$ be

$$t_\star(\lambda)=\operatorname*{\arg\,\max}_{t\in[0,1]}\left(\mathbb{P}(\Phi<t)-\left(1-\frac{\lambda(1-\pi_0)}{\mathbb{P}(\Phi<t)}\right)\mathbb{E}\left[\Phi\,\mathsf{1}\{\Phi<t\}\right]\right).$$

Limiting FDP and TPP conjecture. The limiting false discovery proportion and true positive proportion of TPoP satisfy

$$\lim_{d\to\infty,\,n/d\to\delta}\operatorname{FDP}(\boldsymbol T_P(\,\cdot\,;t))=\mathbb{P}(\beta_0=0\mid\Phi<t),$$ $$\lim_{d\to\infty,\,n/d\to\delta}\operatorname{TPP}(\boldsymbol T_P(\,\cdot\,;t))=\mathbb{P}(\Phi<t\mid\beta_0\neq0),$$

and the limiting false discovery proportion and true positive proportion of CPoP satisfy

$$\lim_{d\to\infty,\,n/d\to\delta}\operatorname{FDP}(\boldsymbol C_P(\,\cdot\,;\lambda))=\mathbb{P}(\beta_0=0\mid\Phi<t_\star(\lambda)),$$ $$\lim_{d\to\infty,\,n/d\to\delta}\operatorname{TPP}(\boldsymbol C_P(\,\cdot\,;\lambda))=\mathbb{P}(\Phi<t_\star(\lambda)\mid\beta_0\neq0).$$

These formulas are suggested by a heuristic replica calculation linking the joint empirical distribution of $(\beta_{0,j},P_j(\mathcal D))$ to the scalar model; proving the conjecture remains open.