In the following post it was shown by mpiktas that the sample correlation of two I(1) series converge to a random variable. On the other, given two cointegrated I(1) variables the OLS estimator is super-consistent. How this two facts settle down together using the following relation between the OLS estimator $\hat {\beta}$ and the correlation?

$\hat {\beta} = {\rm cor}(Y_i, X_i) \cdot \frac{ {\rm SD}(Y_i) }{ {\rm SD}(X_i) }$

Thanks