There's a quick way to tell the difference between coordinate adjectives and cumulative adjectives. If you can rearrange the adjectives without changing the meaning of the phrase, you've got coordinate adjectives, and you must use commas. If you can't rearrange them without changing the meaning of the phrase, you've got cumulative adjectives, and you shouldn't use commas.
Grammar Girl has a convenient set of examples that I'll reproduce here:
Coordinate adjectives are adjectives in a row that each separately modify the noun that follows (1), as in “heavy, bulky box.” Both “heavy” and “bulky” modify “box.” You can even rearrange the adjectives and say, “bulky, heavy box.”
"Bulky, heavy box" and "heavy, bulky box" mean the same thing as each other: the box is both bulky and heavy.
In the phrase “exquisite custom houseboat,” “custom” modifies “houseboat”—they become a unit—and then “exquisite” modifies “custom houseboat.”
To say "exquisite custom houseboat" means that you want to buy a custom houseboat, and to make sure that it's exquisite. If you reversed them - "custom exquisite houseboat" - it would mean that your first thought is for buying an exquisite houseboat, and then also you want it to be custom.
That is what is happening in your sentence. The speakers don't just want to reach a solution that is comprehensive, long-term, and mutually-agreed. They want to reach:
- a [comprehensive solution] that is long-term and mutually-agreed
- a [long-term comprehensive solution] that is mutually-agreed
- a [mutually-agreed long-term comprehensive solution]
I'm not sure if that way of breaking it down makes sense or not. Basically, what happens is the adjective closest to the noun sort of fuses with it to become one noun-unit that the other adjective describes. But then that happens again with the next adjective - it fuses with the adj-noun unit to become a superunit, which the third adjective finally describes.
So you don't need the commas because "long-term comprehensive solution" is one "unit", which "mutually-agreed" is describing. It's the same as how you don't need a comma in "the fuzzy green tennis ball" (or "the exquisite custom houseboat").