Conjecture 1.1 of Chen–Hussain–Yau–Zuo on generalized moduli algebras

Let $(V,0)$ be an isolated complex hypersurface singularity defined by $f\in\mathbb{C}\{x_1,\ldots,x_n\}$, with $n\ge 2$ and $\operatorname{mult}(f)\ge 3$. Let $A(V)$ be the moduli (Tjurina) algebra and $A^*(V)$ the generalized moduli algebra, and define $L(V)=\operatorname{Der}_{\mathbb{C}}(A(V),A(V))$ and $L^*(V)=\operatorname{Der}_{\mathbb{C}}(A^*(V),A^*(V))$. Then $\dim_{\mathbb{C}}L^*(V)=\dim_{\mathbb{C}}L(V)$, equivalently $\lambda^*(V)=\lambda(V)$, where $\lambda(V)=\dim_{\mathbb{C}}L(V)$ and $\lambda^*(V)=\dim_{\mathbb{C}}L^*(V)$.