Merge Meeting Intervals

You are given a list of meeting intervals and a non-negative integer called gap. Each meeting interval is represented as a pair of integers [start, end] where start ≤ end, and these integers indicate time (e.g., minutes since midnight). The goal is to merge all intervals that either overlap or are separated by a gap less than or equal to the specified gap.

Your task:

• Write a function that takes in the list of intervals and the gap value.
• Merge intervals if the gap between the end of one interval and the start of the next is less than or equal to the given gap. Two intervals [a, b] and [c, d] can be merged if c - b ≤ gap.
• Return the resulting list of merged intervals sorted in ascending order by start time.

Constraints and notes:

• The input list might not be sorted.
• Ensure your solution handles edge cases such as an empty list or intervals that only barely miss overlapping by exactly the gap value.
• You may assume all numeric values are non-negative integers.

Write your solution in the programming language of your choice. Focus on correctness and clarity.