Understanding 5 consecutive "that"s in this sentence

(This sentence was told as an entertainment by my English teacher 8 years ago.) She presented us with the following sentence:

She said that that that that that he said was wrong.

I had a bit of trouble trying to figure out what each that means, and am still having a bit of trouble even now.

After killing my brain for hours on this, I worked this out:

• There are only 3 predicate verbs in the sentence, so there should be exactly 2 that's that start a subclause
• No two consecutive that's can have a solid meaning, or it's a grammar error
• Given the above two points, the second and the fourth that's are starts of two subclauses, and the first, the third and the last thats have a solid meaning.
• The first that and the whole subclause led by the second that are appositive. Similarly, the third that and the subclause led by the fourth that are appositive.

The sentence now looks like this:

She said *that* that *that* that *that* he said was wrong.
|             |      |           |            |         |
|             |      | inner - 1 ~~~~~~~~~~~~~~         |
|             |    2 ~~~~~~.....................~~~~~~~~~
~~~~~~~~~~~~~~~ 3 - outer

It's then that problems arise. I can't understand why the last that is fine here. If the above ideas are correct, then the third that is the object of the innermost subclause (marked as 1), and the last that doesn't serve as any structural element in that clause.

What's the role of the last that in the innermost subclause? Or did I interpret the sentence wrongly?

I think it's meant to be interpreted this way:

She said that that "that that" that he said was wrong.

(So, "that that" often appears in English; it is tricky to use correctly. Some student, John, used a "that that". John did so badly. Hence: the teacher said that that "that that" that John used was wrong.)

Once you emphasize the "that that" originally spoken item, it is fairly clear.

So just TBC, your diagram is incorrect and not solvable: the two thats I show grouped, are grouped.

(As mentioned in a comment, for me it makes more sense when the "thing" in italics is just "that". That's the way I've usually heard it posed. But yes, a that that is enough of a thing that you can, uh, make it the thing in the trick sentence!)