A connector housing assembly is disclosed having a housing, a plurality of compressible heat conducting members and a monolithic heat sink. The housing has a first direction corresponding to a height direction of the housing, a second direction parallel to a heat generating module insertion direction, and a third direction perpendicular to the first direction and the second direction. The housing includes a plurality of module receiving ports positioned side by side in the third direction, a plurality of conductor receiving spaces corresponding to the plurality of module receiving ports respectively, and a heat sink installation surface. The plurality of compressible heat conducting members are positioned in the plurality of conductor receiving spaces respectively, each having an elasticity in the first direction. The monolithic heat sink is positioned on a heat sink installation surface of the housing, covering and in surface-contact with the plurality of compressible heat conducting members.