For my question, let us consider the following scenario.
We have a quasi-variety $mathcal{A}$ generated by a finite algebra $mathbf{M}$ (i.e. $mathcal{A} = mathbb{ISP}(mathbf{M})$). Now, let $mathbf{A}$ be another finite algebra in $mathcal{A}$. Assume that there exists an integer $k$ such that, for each $n in mathbb{N}$, every homomorphism $f colon mathbf{A}^n rightarrow mathbf{M}$ is essentially at most $k$-ary.
My question is: Is it guaranteed that there exists a finite algebra $mathbf{M'} in mathcal{A}$ with $mathbf{A} in mathbb{ISP}(textbf{M'})$ such that, for each $n in mathbb{N}$, every homomorphism $f colon mathbf{A}^n rightarrow mathbf{M'}$ is essentially at most unary?
To illustrate this question, I will now give an example in which this is true. Let $textbf{M}$ and $textbf{A}$ be two finite distributive lattices. Now, there exists an integer $k$ such that the essential artiy of every $f colon mathbf{A}^n rightarrow mathbf{M}$ is at most $k$-ary. We can define $mathbf{M'}$ to be the (up to isomorphism unique) two-element distributive lattice.
Note that $mathbb{ISP}(mathbf{M'}) = mathbb{ISP}(mathbf{M})$ is not required (although it is true in the example). It is only required to have $mathbf{A} in mathbb{ISP}(mathbf{M'})$.
Is this always possible? I believe it is not, but I do not know any counterexample up to this point in time. Does anybody know a counterexample, a proof or has a feeling of whether this statement is true or false?
Note: If the answer to my question would be positive, this would have some very nice consequences for centralizer clones (which is my main research interest at the time). In fact, this would mean (and I can explain later, why that is) that every centralizer clones that has bounded essential arity cannot contain many types of functions. Among them: Near-unanimity operations, minority operations, semiprojections etc.. Furthermore, it would lead to a full characterization of all minimal clones in centralizer clones with bounded essential arity.
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