Strictness of the Dimension Inequality for Zeroth de Rham Cohomology

Determine whether the inequality \(\dim_K H^0_{\mathrm{dR}}(A)\le 1+\dim_K H^1(\ker\Phi)\) can be strict for a local Artinian \(K\)-algebra \(A=P/I\), where \(P=K[x_1,\dots,x_n]\), \(\Phi:\Omega^\bullet_{\operatorname{gr}(A)}\to\operatorname{gr}(\Omega_A^\bullet)\) is the canonical filtered differential graded algebra homomorphism, and \(K\) is the base field.

Background

Theorem \ref{thm:ZerothDeRham} establishes the upper bound $$\dim_K H^0_{\mathrm{dR}}(A)\le 1+\dim_K H^1(\ker\Phi)$$, where $H^0_{\mathrm{dR}}(A)=\ker(d_A)$ and $\Phi$ compares the de Rham complex of the associated graded ring with the associated graded de Rham complex of $A$. The authors report that they found no example in which this inequality is strict, but also no proof that equality always holds. The unresolved task is therefore to decide whether strict inequality can occur for any local Artinian algebra in the stated setting.