A finite reduced Laver-like algebra is a finite algebra $(X,*,1)$ that satisfies the identities $1*x=x,x*1=1,x*(y*z)=(x*y)*(x*z)$ and where there is a natural number $n$ and a function $\mathrm{crit}:X\rightarrow n+1$ where
$\mathrm{crit}(x)=n$ if and only if $x=1$,
$\mathrm{crit}(x*y)=\mathrm{crit}(y)$ whenever $\mathrm{crit}(y)<\mathrm{crit}(x)$, and
$\mathrm{crit}(x*y)>\mathrm{crit}(y)$ whenever $\mathrm{crit}(x)\leq\mathrm{crit}(y)<n$.
We say that a finite reduced Laver-like algebra $(X,*,1)$ is critically simple if whenever $\simeq$ is a congruence on $(X,*,1)$ where there are $x,y\in X$ with $x\simeq y,x\neq y$, there is some $z\in X\setminus\{1\}$ with $z\simeq 1$.
We say that $(X,*,1)$ is critically subsimple if $x,y\in X,x*x=1,y*y=1,\mathrm{crit}(x)=\mathrm{crit}(y)$ implies that $x=y$ for each $x,y\in X$. Every critically simple finite reduced Laver-like algebra is critically subsimple.
Suppose that $(X,*,1)$ is a critically subsimple finite reduced Laver-like algebra generated by $(x_{a})_{a\in A}$ where $A$ is finite. Then does there exist a critically simple finite reduced Laver-like algebra $(Y,*,1)$ generated by $(y_{a})_{a\in A}$ along with a homomorphism $\phi:Y\rightarrow X$ such that $\phi(y_{a})=x_{a}$ for each $a\in A$ and an element $c\in Y$ such that if $y_{1},y_{2}\in Y$, then $\phi(y_{1})=\phi(y_{2})$ if and only if $c*y_{1}=c*y_{2}$?