A pull-in head for a high voltage cable provided with a multi-wire conductor, the end portion of the multi-wire conductor being arranged in a stepped manner, the multi-wire conductor being arranged tapering, with a layer-by-layer exposure of the wires in the direction towards the end face of the multi-wire conductor, wherein the pull-in head comprises: a conductor clamp assembly comprising a plurality of clamping segments, wherein each clamping segment has an axial first central channel configured to receive the multi-wire conductor, wherein at least some of the clamping segments have mutually differing central channel width and central channel depth dimensions, the clamping segments being configured to be arranged one after the other in an order with a decreasing central channel width and central channel depth dimension, whereby the first central channels together form a multi-wire conductor channel which is tapering in an axial direction, and wherein each clamping segment is configured to provide individual clamping of the multi-wire conductor, wherein the clamping segments have central channel width and depth dimensions that are adapted to the tapering stepped configuration of the end portion of the multi-wire conductor, wherein the conductor clamp assembly includes a support structure, wherein the clamping segments are configured to be assembled with the support structure to thereby clamp the multi-wire conductor between the clamping segments and the support structure, wherein the support structure has an inner surface configured to face the clamping segments, which inner surface is provided with an axial second central channel having a stepped tapering structure in the axial direction of the conductor clamp assembly, the axial second central channel having a tapering structure adapted to the tapering stepped configuration of the end portion of the multi-wire conductor, wherein the axial second central channel and the axial first central channels face each other when the clamping segments are assembled with the support structure, whereby a stepped tapering central common channel is formed between the axial second central channel and the axial first central channels for clamping the end portion of the multi-wire conductor, and a hollow lifting head configured to receive the conductor clamp assembly for providing the lifting head around the conductor clamp assembly. a conductor clamp assembly comprising a plurality of clamping segments, wherein each clamping segment has an axial first central channel configured to receive the multi-wire conductor, wherein at least some of the clamping segments have mutually differing central channel width and central channel depth dimensions, the clamping segments being configured to be arranged one after the other in an order with a decreasing central channel width and central channel depth dimension, whereby the first central channels together form a multi-wire conductor channel which is tapering in an axial direction, and wherein each clamping segment is configured to provide individual clamping of the multi-wire conductor, wherein the clamping segments have central channel width and depth dimensions that are adapted to the tapering stepped configuration of the end portion of the multi-wire conductor, wherein the conductor clamp assembly includes a support structure, wherein the clamping segments are configured to be assembled with the support structure to thereby clamp the multi-wire conductor between the clamping segments and the support structure, wherein the support structure has an inner surface configured to face the clamping segments, which inner surface is provided with an axial second central channel having a stepped tapering structure in the axial direction of the conductor clamp assembly, the axial second central channel having a tapering structure adapted to the tapering stepped configuration of the end portion of the multi-wire conductor, wherein the axial second central channel and the axial first central channels face each other when the clamping segments are assembled with the support structure, whereby a stepped tapering central common channel is formed between the axial second central channel and the axial first central channels for clamping the end portion of the multi-wire conductor, and a hollow lifting head configured to receive the conductor clamp assembly for providing the lifting head around the conductor clamp assembly.