Let $U_{r,l,q}$ be the set of finite dimensional local algebras $A$ over a finite field with $q$ elements such that $A/J$ is $r$-dimensional for a number $r \geq 2$ and such that $J^l=0$ for the Jacobson radical $J$ (=unique maximal ideal) of $A$ for some $l \geq 2$. Let $W_{r,l,q}$ be the subset of $U_{r,l,q}$ consisting of those algebras that are additionally Frobenius algebras (or equivalently Gorenstein algebras). Let $u_{r,l,q}$ be the cardinality of $U_{r,l,q}$ and $w_{r,l,q}$ the cardinality of $W_{r,l,q}$. Question:

Is it true that $\frac{w_{r,l,q}}{u_{r,l,q}}$ goes to zero for $l \rightarrow \infty $ for any $r,q$? Can one even give asymptotics for spcific values of $r$ and $q$?

What if we restrict to commutative algebras?

Let $U_{r,l,q}$ be the set of finite dimensional local algebras $A$ over a finite field with $q$ elements such that $J/J^2$ is $r$-dimensional for a number $r \geq 2$ and such that $J^l=0$ for the Jacobson radical $J$ (=unique maximal ideal) of $A$ for some $l \geq 2$. Let $W_{r,l,q}$ be the subset of $U_{r,l,q}$ consisting of those algebras that are additionally Frobenius algebras (or equivalently Gorenstein algebras). Let $u_{r,l,q}$ be the cardinality of $U_{r,l,q}$ and $w_{r,l,q}$ the cardinality of $W_{r,l,q}$. Question:

Is it true that $\frac{w_{r,l,q}}{u_{r,l,q}}$ goes to zero for $l \rightarrow \infty $ for any $r,q$? Can one even give asymptotics for spcific values of $r$ and $q$?

What if we restrict to commutative algebras?