Solution conductance is determined by the concentration of ions. Upon dissolution of coordination complexes in water, counter-ions dissociate and contribute to conductance. Ions within the coordination sphere remain undissociated.
Analysis of each complex:
1. $[Co(NH_3)_4Cl_2]Cl$ yields $[Co(NH_3)_4Cl_2]^+$ and $Cl^-$, resulting in 2 ions.
2. $[Co(NH_3)_6]Cl_3$ yields $[Co(NH_3)_6]^{3+}$ and $3Cl^-$, resulting in 4 ions.
3. $[Co(NH_3)_5Cl]Cl_2$ yields $[Co(NH_3)_5Cl]^{2+}$ and $2Cl^-$, resulting in 3 ions.
4. $[Co(NH_3)_3Cl_3]$ lacks external counter-ions. Consequently, it does not dissociate into ions in solution, existing as a neutral molecule. As such, it is a non-electrolyte with minimal conductance.
The complex exhibiting minimum conductance will be the one that generates the fewest ions in solution.
Therefore, the compound with the minimum conductance is $[Co(NH_3)_3Cl_3]$.