If you allow negative integers, you get negative exponents. To accomodate negative exponents, you can expand the domain again, to rational numbers. If you allow rational numbers, then you have to allow rational exponents, which means you can take arbitrary nth roots. Which you could do, but now you're most of the way to the complex numbers (plus roots aren't unique anymore).
Why was it important to Tarski to limit the domain to positive integers? That seems pointlessly arbitrary.
If you allow negative integers, you get negative exponents. To accomodate negative exponents, you can expand the domain again, to rational numbers. If you allow rational numbers, then you have to allow rational exponents, which means you can take arbitrary nth roots. Which you could do, but now you're most of the way to the complex numbers (plus roots aren't unique anymore).
Truly, you could say that about many conjectures, especially more “fun” classical ones.