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.

Example 1:

    Input: num = "52"
    Output: "5"
    Explanation: The only non-empty substrings are "5", "2", and "52". "5" is the only odd number.

Example 2:

    Input: num = "4206"
    Output: ""
    Explanation: There are no odd numbers in "4206".

Example 3:

    Input: num = "35427"
    Output: "35427"
    Explanation: "35427" is already an odd number.

Constraints:

- `1 <= num.length <= 105`
- `num` only consists of digits and does not contain any leading zeros.