The Buckley-James estimator (BJE) of the parameter, under the semi-parametric linear regression model with right-censored data, was initially proposed in 1979 and has since become the standard extension of the least squares estimator. The consistency and efficiency of BJE are contingent upon various regularity conditions, including the identifiability condition. However, it remains unclear whether these conditions also guarantee the existence of BJE. In their original work, Buckley and James presented an iterative algorithm to find a solution for BJE while acknowledging that it may not be unique or even exist at all. Kong and Yu (2007) claimed to have identified a case where BJE does not exist; however, subsequent research has shown this claim to be incorrect. Consequently, the question regarding the general existence of BJE still remains unresolved. In our paper, this open problem is settled.