Sure, I can help you create a test case generator for the problem "D: Rescue a Postal Worker". The test case generator will create random inputs that conform to the problem's constraints. Here's the Python code for it:

```python
import random

def generate_test_case():
    # Number of vertices
    n = random.randint(3, 1000)
    # Number of edges
    m = random.randint(1, min(2000, n*(n-1)//2))
    # Number of dropped mails
    k = random.randint(1, 6)
    # Starting vertex
    p = random.randint(1, n)
    
    edges = set()
    for _ in range(m):
        while True:
            x = random.randint(1, n)
            y = random.randint(1, n)
            if x != y and (x, y) not in edges and (y, x) not in edges:
                w = random.randint(1, 1000)
                edges.add((x, y, w))
                break
    
    mail_positions = set()
    while len(mail_positions) < k:
        s = random.randint(1, n)
        t = random.randint(1, n)
        if s != t and s != p and (s, t) not in mail_positions:
            mail_positions.add((s, t))
    
    test_case = f"{n} {m} {k} {p}\n"
    for x, y, w in edges:
        test_case += f"{x} {y} {w}\n"
    for s, t in mail_positions:
        test_case += f"{s} {t}\n"
    
    return test_case

# Example usage
print(generate_test_case())
```

### Explanation

1. **Vertices and Edges**:
    - `n`: The number of vertices, chosen randomly between 3 and 1000.
    - `m`: The number of edges, chosen randomly between 1 and the minimum of 2000 or the maximum number of possible edges for `n`.

2. **Starting Vertex and Mail**:
    - `p`: The starting vertex, chosen randomly between 1 and `n`.
    - `k`: The number of dropped mails, chosen randomly between 1 and 6.

3. **Edges Generation**:
    - Each edge is chosen randomly, ensuring no self-loops and no multiple edges between the same vertices. The weight of each edge is between 1 and 1000.

4. **Mail Drops**:
    - The positions of the dropped mails and their destinations are chosen randomly, ensuring that no mail drops at the starting vertex and each mail has a unique source-destination pair.

5. **Test Case Construction**:
    - The test case is formatted as the problem specifies, combining the vertices, edges, and mail drops into a single string.

This function will generate a random test case whenever it is called. You can adjust the random seed or modify the constraints to generate specific types of test cases if needed.