On the J\'onsson distributivity spectrum

Suppose throughout that $\mathcal V$ is a congruence distributive variety. If $m \geq 1$, let $J _{ \mathcal V} (m)$ be the smallest natural number $k$ such that the congruence identity $\alpha ( \beta \circ \gamma \circ \beta \dots ) \subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta \circ \dots$ holds in $\mathcal V$, with $m$ occurrences of $\circ$ on the left and $k$ occurrences of $\circ$ on the right. We show that if $J _{ \mathcal V} (m) =k$, then $J _{ \mathcal V} (m \ell ) \leq k \ell$, for every natural number $\ell$. 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 $J _{ \mathcal V} (m) \leq m$, for every $m \geq 3$ (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.