To solve the given question, we need to understand the concepts of gravitational potential and gravitational field.

The gravitational potential at a point in a field is defined as the work done per unit mass in bringing a mass from infinity to that point. It is a scalar quantity and is the same at points $A$ and $B$ as per the given condition.

On the other hand, the gravitational field at a point is a vector quantity and represents the force experienced by a unit mass placed at that point. The problem states that the gravitational fields at points $A$ and $B$ are different.

Let's analyze the correct option:

1. No work is done on an object by a conservative force (like gravity) when moving between two points of equal potential. This is because the work done by a conservative force is equal to the negative change in potential energy. Since the potential is the same at both points, the change in potential energy is zero, and hence, no work is done.

Let's eliminate the incorrect options:

1. Positive work is done by gravity in moving the object from point $A$ to point $B$: This is incorrect because no work is done if the gravitational potential is the same at both points.
2. Negative work is done by gravity in moving the object from point $A$ to point $B$: This is also incorrect, as explained above, no work is done.
3. Gravity must be non-conservative in such a region: This statement is incorrect because gravitational force is always conservative regardless of the field strength at different points.

Based on the reasoning above, the correct answer is:

No work is done by gravity in moving the object from point $A$ to point $B$.