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Existing works in optimal filtering for linear Gaussian systems with arbitrary unknown inputs assume perfect knowledge of the noise covariances in the filter design. This is impractical and raises the question of whether and under what conditions one can identify the noise covariances of linear Gaussian systems with arbitrary unknown inputs. This paper considers the above identifiability question using the correlation-based autocovariance least-squares (ALS) approach. In particular, for the ALS framework, we prove that (i) the process noise covariance Q and the measurement noise covariance R cannot be uniquely jointly identified; (ii) neither Q nor R is uniquely identifiable, when the other is known. This not only helps us to have a better understanding of the applicability of existing filtering frameworks under unknown inputs (since almost all of them require perfect knowledge of the noise covariances) but also calls for further investigation of alternative and more viable noise covariance methods under unknown inputs. Especially, it remains to be explored whether the noise covariances are uniquely identifiable using other correlation-based methods. We are also interested to use regularization for noise covariance estimation under unknown inputs, and investigate the relevant property guarantees for the covariance estimates. The above topics are the main subject of our current and future work.