The extension to every modulus is false as stated. In ℤ/4ℤ, (3, 0, 1) is a progression with distinct terms.
Why it breaks
The common difference is 1, so 3 + 1 = 0 and 0 + 1 = 1 modulo 4.
Scope
This refutes the direct all-moduli claim. It does not rule out a modified construction for selected moduli.