Convert all Unix Time in a file to human date

I want to replace all the UNIX time in a a JSON data file with the same UNIX time followed by a human readable date format.

For example, I want to replace "1234567890123456" (time in microsecond) with "1234567890123456,2009-02-14,00:31:30,sat".

I tried with

sed -r 's/([0-9]{16})/\1'"`date -d @ \1`"'/g'

but it doesn't work. I'm new in UNIX world, can you suggest me a solution? Thank you so much.

Cordiality

I attach a sample of my JSON file:

{
  "example": [
    {
      "query": {
        "text": "string 1",
        "id": [
          {
            "timestamp": "1234567890123456"
          }
        ]
      }
    },
    {
      "query": {
        "text": "string 2",
        "id": [
          {
            "timestamp": "1234567891123456"
          }
        ]
      }
    },
    {
      "query": {
        "text": "string 3",
        "id": [
          {
            "timestamp": "1234567893123456"
          }
        ]
      }
    }
  ]
}

Copyright License：
Author:「Jean S.」,Reproduced under the CC 4.0 BY-SA copyright license with link to original source & disclaimer.
Link to：https://stackoverflow.com/questions/30151378/convert-all-unix-time-in-a-file-to-human-date