Universality of the likelihood-ratio statistic under stable finite-order VAR dynamics

Fix $k\in\mathbb{N}$ and $C>0$. Suppose that the data-generating process satisfies

$$\Delta X_t=\mu+\sum_{i=1}^{k-1}\Gamma_i\Delta X_{t-i}+\varepsilon_t,\qquad t=1,\ldots,T,$$

where the innovations are independent and identically distributed as $\mathcal{N}(0,\Lambda)$, the covariance matrix satisfies $\|\Lambda\|_2<C$ and $\|\Lambda^{-1}\|_2<C$, $\|\Gamma_i\|_2<C$ and $\operatorname{rank}(\Gamma_i)<C$ for $1\leq i\leq k-1$, all roots of

$$\det\left(\mathbf{1}_N-\sum_{i=1}^{k-1}\Gamma_i z^i\right)=0$$

are such that $|z|>1+C^{-1}$, and $\|\Gamma_j\Delta X_{1-i}\|_2\leq C$ and $\|\Gamma_j\mu\|\leq C$ for $1\leq i,j\leq k-1$. Universality conjecture. As $T,N\to\infty$ with $T/N\in[k+1+C^{-1},C]$, the conclusion of Theorem J continues to hold with the same centering and scaling constants $c_1(N,T)$ and $c_2(N,T)$:

$$\frac{\sum_{i=1}^{r}\ln(1-\tilde{\lambda}_i)-r\,c_1(N,T)}{N^{-2/3}c_2(N,T)}\xrightarrow[T,N\to\infty]{d}\sum_{i=1}^{r}\mathfrak{a}_i.$$

This extends the asserted asymptotics beyond the null model to stable finite-order vector autoregressions with bounded coefficient ranks and covariance conditioning. The supplied text does not identify whether this conjecture is proved or remains open.