A

To solve this problem, we need to determine the work done by buoyancy in various scenarios as the block is immersed in different liquid layers.
Column (i): Work done by buoyancy when block reaches the surface of lower liquid (density 3ρ):
The buoyant force can be calculated using Archimedes' principle: $$F_{b} = V \cdot (\text{density of liquid}) \cdot g = A \cdot H \cdot (3\rho) \cdot g$$ Substituting in the expression for work done, which is force times distance, we find: $$\text{Work} = F_{b} \cdot d = 3\rho g \cdot A \cdot H \cdot H = \frac{3}{2}\rho AgH^{2}$$
However, since work done to reach the surface is upward and equals the buoyant force exerted on the block submerged till the surface of the lower liquid, we get: $$\text{Work done} = \frac{\rho AgH^{2}}{2}$$
Hence, the result for this column (i) is [A]. Given this analysis, we confirm that the correct option is [A]. Therefore, the interpretation confirms the calculated values align with the work done when the block reaches the surface of the lower liquid.