Please solve the programming task below using a self-contained code snippet in a markdown code block.

You are given a string num, representing a large integer. Return the largest-valued odd integer (as a string) that is a non-empty substring of num, or an empty string "" if no odd integer exists.
A substring is a contiguous sequence of characters within a string.
 
Please complete the following python code precisely:
```python
class Solution:
    def largestOddNumber(self, num: str) -> str:
```