Whether LKJ prior is proper

I want to use the LKJ prior for modeling covariance matrix. I wonder whether this is a proper prior.

The LKJ distribution is a proper prior over a correlation matrix. If a covariance matrix is expressed as the transformed parameter \boldsymbol{\Sigma} = \boldsymbol{\Delta} \mathbf{L} \mathbf{L}^\top \boldsymbol{\Delta} where \boldsymbol{\Delta} is a diagonal matrix of standard deviations and \mathbf{L} is a Cholesky factor of a correlation matrix, then the distribution of \boldsymbol{\Sigma} before seeing the data is proper if the diagonal elements of \boldsymbol{\Delta} all have proper priors and the LKJ distribution is used for \mathbf{L} \mathbf{L}^\top.