Exact simultaneous and sequential limits for the scaled LR statistic

Let $LR_{0,p,T}$ be Johansen's likelihood-ratio statistic for testing no cointegrating relationships, with $p,T\to\infty$ and $p/T\to c<1/2$ in the simultaneous regime. Scaled LR limit conjecture. The simultaneous and sequential lower bounds are the corresponding limits, namely

$$\lim_{p,T\to_c\infty}\frac{1}{2p^{2}}LR_{0,p,T}=\frac{1+c}{2c^{2}}\ln(1+c)-\frac{1-c}{2c^{2}}\ln(1-c)+\frac{1-2c}{2c^{2}}\ln(1-2c),$$

and

$$\operatorname*{plim}_{p\to\infty}\lim_{T\to\infty}\frac{1}{2p^{2}}LR_{0,p,T}=1.$$

These identities would upgrade the paper's simultaneous and sequential lower bounds to exact asymptotic limits.