Davenport constant equality for multiplicative semigroups of polynomial quotient rings

Let $p>2$ be a prime, let $f(x)\in \mathbb{F}_p[x]$ be a non-constant polynomial, and let $\mathcal{S}_{f(x)}^p$ denote the multiplicative semigroup of the quotient ring $\mathbb{F}_p[x]/\langle f(x)\rangle$. Let $U(\mathcal{S}_{f(x)}^p)$ denote its group of units. Davenport constant equality conjecture. One has

$$D(\mathcal{S}_{f(x)}^p)=D\bigl(U(\mathcal{S}_{f(x)}^p)\bigr).$$

The paper proves this equality when $f(x)$ factors into pairwise non-associated irreducible polynomials, and verifies the special case $f(x)=(x+1)^2$; the general case for non-constant $f(x)$ remains open.