Let $\mathbb{R}[x, y]_d$ denote the vector space of homomogeneous polynomials in the indeterminates $x, y$ of degree $d$ with real coefficients. As a graded ring $A :=\mathbb{R}[x, y] = \bigoplus_{d = 0}^\infty \mathbb{R}[x, y]_d$ acts on the graded vector space $M := A(1)_{\ge 0} = \bigoplus_{k = 0}^\infty \mathbb{R}[x, y]_{k + 1}$ by multiplication: $f \otimes g \mapsto fg: \mathbb{R}[x, y]_d \otimes \mathbb{R}[x, y]_{k + 1} \to \mathbb{R}[x, y]_{d + k + 1}$.
Fix a graded ring homomorphism $\varphi : \mathbb{R}[x, y] \to \mathbb{R}[t]$.
Suppose that $\psi : M \to \mathbb{R}[t]$ is a graded linear map satisfying the property: $$\psi(f g) = \varphi(f) \psi(g) \quad \quad \forall f\in \mathbb{R}[x, y], \forall g \in M.$$ For $k \ge 0$, given any $g = \sum_{i = 0}^k c_{i} x^i y^{k - i} \in \mathbb{R}[x, y]_{k + 1} = M_k$, is it true that $\psi(g) = \sum_{i = 0}^k c_i (ax)^i (by)^{k - i} t^k$ for some real numbers $a, b$ such that $\varphi(bx - ay) = 0$?