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The aim of this paper is to extend the theoretical framework of the P * (κ) horizontal linear complementarity problem over Cartesian product of symmetric cones (Cartesian P * (κ)-SCHLCP). The concepts of column and row sufficiency are defined for a linear operator on a Euclidean Jordan algebra. Some connections between the P * (κ) property of a linear operator on a Euclidean Jordan algebra and its P 0 property as well as its column sufficiency are presented. Then these definitions and connections are generalized to a pair of linear operators determining the Cartesian P * (κ)-SCHLCP. It is also demonstrated that the Cartesian P * (κ)-SCHLCP can be equivalent with a P * (κ) linear complementarity problem over Cartesian product of symmetric cones (Cartesian P * (κ)-SCLCP), in a certain sense. Finally, based on some obtained results and using a suitable potential function, the main task of the paper is done, which is to show that the central trajectory of the Cartesian P * (κ)-SCHLCP exists and is unique. Therefore, we open the way for presenting and extending the interior-point methods for the Cartesian P * (κ)-SCHLCP.