Unit-group equality for irreducible polynomials

Let $k$ be the ground field, let $f$ be the polynomial defining the affine variety in the paper, and let $T$ be its coordinate ring. Write $T^\ast$ for the group of units of $T$. Unit-group conjecture. If $f$ is irreducible, then

$$T^\ast=k^\ast.$$

The paper gives partial answers in Theorems 2.10 and 2.12; the general assertion is presented as an unresolved conjectural statement.