covariance of coefficient estimates in a linear regression model

Question

paper

I am confused about how the above paper gets the correlation of the coefficient estimates $Cor^2(\hat{\gamma_j}, \hat{\gamma_k})$. The covariance matrix of the coefficient would be $(Z^TZ)^{-1}\sigma^2$. If the $Z_j$ and $Z_k$ are independent, so the diagonal of this matrix would be 0. But how the author can get the covariance not being 0. If this covariance are because they use the same $X$, I tried to calculate the covariance like this $cov(\hat{\gamma_j}, \hat{\gamma_k})=cov(\frac{\widehat{\operatorname{Cov}}_{n_X}\left(X, Z_j\right)}{\widehat{\operatorname{Cov}}_{n_X}\left(Z_j, Z_j\right)}, \frac{\widehat{\operatorname{Cov}}_{n_X}\left(X, Z_k\right)}{\widehat{\operatorname{Cov}}_{n_X}\left(Z_k, Z_k\right)})$ as in the picture below. The result formula is really long, which I don't think is right. Does anyone know how should I get the formula as in the paper, and why the two ways I tried are wrong? Thanks.

proof