So I checked up on it, and no, it is not the case. Also, I have some bad news. If we assume the Schroder functions (accordingly, the Abel functions) are iterated exponentials, then that means that the associated hyperoperation hierarchy would have the recursion rule:

which (if n=2 is multiplication) requires that (n=1) is , which is not addition, so this scheme does not form a hyperoperation hierarchy. However, if I might draw your attention to the Tetration Ref. section 2.3.4, the recursion rule:

has many of the same characteristics, and satisfies addition (n=1), mul (n=2), and powers (n=3). The only difference is in the heighest exponent.

which (if n=2 is multiplication) requires that (n=1) is , which is not addition, so this scheme does not form a hyperoperation hierarchy. However, if I might draw your attention to the Tetration Ref. section 2.3.4, the recursion rule:

has many of the same characteristics, and satisfies addition (n=1), mul (n=2), and powers (n=3). The only difference is in the heighest exponent.