Christophersen's problem on automorphism groups of local algebras

Let $A=\mathbb{C}\oplus\mathfrak{m}$ be a finite-dimensional local algebra of dimension $n$, where $\mathfrak{m}$ is its maximal ideal, and let $\operatorname{Aut}(A)^{\circ}$ denote the identity component of its algebraic automorphism group. Christophersen's problem. For every local algebra $A$, one has

$$\dim\operatorname{Aut}(A)^{\circ}\geq n-1.$$

Furthermore, if

$$\dim\operatorname{Aut}(A)^{\circ}=n-1,$$

then $A$ is isomorphic to $\mathbb{C}[t]/(t^n)$. The classification of finite-dimensional local algebras is difficult: infinitely many isomorphism classes occur from dimension seven onward, while the truncated polynomial algebra provides an example attaining the proposed lower bound.