Diamonds have a low work function but are not conductors.
Certainly diamonds such as natural type IIa, across specific planes / axes, are among the world's best electrical insulators, shared with some varieties of polystyrene and perhaps ultraclean PTFE. Low electron emission work function is often identified with high bulk electrical resistivity.
But, thermally diamonds can be the worlds best conductors:
Thermal conductivity, especially in specific orientations is closely associated with effective electron mass. Apparently reprising aspects of W-L theory.