On the J\'onsson distributivity spectrum

Suppose throughout that $\mathcal{V}$ is a congruence distributive variety. If $m\ge 1$, let $J_{\mathcal{V}}(m)$ be the smallest natural number $k$ such that the congruence identity $\alpha ( \beta \circ \gamma \circ \beta \dots ) \subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta \circ \dots $holdsin$\mathcal V$,with$m$occurrencesof$ \circ$ontheleftand$k$occurrencesof$\circ$ontheright\mathrm{.}Weshowthatif$ J _{ \mathcal V} (m) =k$,then$ J _{ \mathcal V} (m \ell ) \leq k \ell $,foreverynaturalnumber$\ell$. The key to the proof is an identity which, through a variety, is equivalent to the above congruence identity, but involves also reflexive and admissible relations. If $J_{\mathcal{V}}(1)=2$, that is, $\mathcal{V}$ is $3$-distributive, then $J_{\mathcal{V}}(m)\le m$, for every $m\ge 3$ (actually, a more general result is presented which holds even in nondistributive varieties). If $\mathcal{V}$ is $m$-modular, that is, congruence modularity of $\mathcal{V}$ is witnessed by $m+1$ Day terms, then $ J _{ \mathcal V} (2) \leq J _{ \mathcal V} (1) + 2m^2-2m -1 $. Various problems are stated at various places.