Conjecture on MDYPL state evolutions with an intercept

Consider the logistic regression model with intercept

$$\Pr(y_j=1\mid\boldsymbol{x}_j)=\zeta'(\theta_0+\boldsymbol{x}_j^\top\boldsymbol{\beta}_0),\qquad j=1,\ldots,n,$$

where $\theta_0\in\mathbb{R}$, the covariates are independent realizations $\boldsymbol{x}_j\sim\mathcal{N}(\boldsymbol{0}_p,\boldsymbol{\Sigma})$, and $\boldsymbol{\beta}_0^\top\boldsymbol{\Sigma}\boldsymbol{\beta}_0\to\gamma^2$ as $n\to\infty$. Let $\hat{\theta}_0^{\mathrm{DY}}$ and $\hat{\boldsymbol{\beta}}^{\mathrm{DY}}$ be the MDYPL estimates of the intercept and remaining coordinates, respectively. Let $({\mu}_*,{b}_*,{\sigma}_*,\iota_*)$ be a nonsingular solution of the four-equation state-evolution system specified in the source, with parameters $(\alpha,\kappa,\gamma,\theta_0)$. MDYPL intercept conjecture. One has

$$\hat{\theta}_0^{\mathrm{DY}}\overset{p}{\longrightarrow}\iota_*$$

as $n\to\infty$, and the source's asymptotic results for the $z$-statistics and likelihood-ratio statistic hold with $({\mu}_*,{b}_*,{\sigma}_*)$ from that system. The intercept extension is outside the paper's current theory; the conjecture predicts that the state evolutions incorporate the intercept and become a system of four equations in four unknowns, with analogous inference results.