With this formula :

@.{high_name : high_name, sections : sections[*].{name : name, item: [item,items_sub[*].*[]][]}}

You can get this :

{
  "high_name": "test",
  "sections": [
    {
      "name": "section1",
      "item": [
        "string1"
      ]
    },
    {
      "name": "section2",
      "item": [
        "string2",
        "deeper string1"
      ]
    }
  ]
}

As you can see I retrieved the string2 with the deep string1 instead of string1, so please tell me if you made a mistakes in your question or you really want string1