The converse between semigroup property and integer normed pairings for quadratic forms

Let $f:\mathbb{R}^2\to\mathbb{R}$ be an integer quadratic form, meaning that its coefficients are integers. The form $f$ has semigroup property if the product of any two values represented by $f$ is again represented by $f$. An integer normed pairing with respect to $f$ is a bilinear map $s:\mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}^2$ that maps $\mathbb{Z}^2\times\mathbb{Z}^2$ into $\mathbb{Z}^2$ and satisfies

$$f(s(\mathbf{x},\mathbf{y}))=f(\mathbf{x})f(\mathbf{y})$$

for all $\mathbf{x},\mathbf{y}\in\mathbb{R}^2$. Conjecture. If an integer quadratic form has semigroup property, then it admits an integer normed pairing. The claim would establish the converse of the sufficient condition discussed in the surrounding text: every integer quadratic form with semigroup property would arise from such a bilinear multiplicative structure. The source does not provide evidence of resolution, so the conjecture is treated as open.