Given a finite dimensional commutative local $K$-algebra $A$ for a field $K$. Associated to $A$ is its dimension $d_A$ and the Betti-sequence $c_i=dim(Ext_A^i(S,S))$ where $S$ is the unique simple $A$-module.

Question: Are there only finitely many such $K$-algebras $A$ with a fixed dimension $d_A$ and associated sequence $c_i$?

If not, then can one characterise for which sequences there are only finitely many for a fixed dimension?

For example for $c_i=1$ constant and $d_A=n$, the unique ones are $A=K[x]/(x^n)$ as we saw in a previous thread Commutative algebras with modules of small complexity .