A

Step 1: Understand the forces acting on the system.
The tension $T$ in the string needs to counteract both the weight of the water column above the piston and the buoyant force acting on the apparatus.

Step 2: Calculate the hydrostatic pressure at depth $h$. The hydrostatic pressure at the depth is given by the equation:
$$P = ho gh$$
where:
- $P$ is the pressure
- $ ho$ is the density of water
- $g$ is the acceleration due to gravity
- $h$ is the depth of the tube's immersion.

Step 3: Set up the force balance. The force due to the pressure at the depth $h$ acting on the piston (with area $A$) will contribute to the force:
$$F_{pressure} = PA = ho ghA$$
The weight of the water column is equivalent to the force exerted by this pressure.

Step 4: Consider the buoyant force experienced by the piston. The volume of water displaced by the piston is equal to its area $A$ multiplied by the height $h$ of the water column above it.
Thus, the buoyant force can be expressed as:
$$F_{buoyant} = ho g V_{displaced} = ho gAh$$

Step 5: The tension in the string can be expressed as:
$$T = F_{gravity} - F_{buoyant}$$
where $F_{gravity}$ is the total weight acting downward.
This means that as the depth $h$ increases, the hydrostatic pressure increases, causing the tension to also increase. Therefore, the tension in the string is directly proportional to the depth of the tube’s immersion.

Conclusion: As the depth of the tube’s immersion increases, the hydrostatic pressure increases leading to an increase in the tension in the string. Hence, the correct answer is A.