This is my latest attempt to simplify an old problem of mine so much that the simplified problem can actually be answered.

Starting with the generating function for unrestricted partitions:

$$(1+x+x^2+x^3+\ldots)(1+x^2+x^4+x^6+\ldots)(1+x^3+x^6+x^9)\ldots$$

Change some of the plus signs in the leftmost expression in parentheses to minus signs. Is it possible that the resulting series has coefficients all of which are $1$, $-1$, or zero?

I believe that the answer is no, but I'm not convinced by my computer aided search.