Step 1: Picture an unsaturated soil.
Above the water table, soil pores are only partly filled with water; the rest of the space holds air.
Step 2: Think about how water actually moves through this mixture.
Water can only travel along the wetted pore paths, so when more of the pore space is filled with water, that is, the degree of saturation is higher, there are more continuous water paths for flow.
Step 3: Compare with the other factors.
Porosity and thickness are fixed structural properties of the soil, and pressure head alone does not decide how connected the water paths are, so it is the degree of saturation that really controls unsaturated hydraulic conductivity.
\[ \boxed{\text{Degree of saturation}} \]