Almost-linear Zarankiewicz bounds in $1$-semi-equational theories

We study multipartite hypergraphs definable in $1$-semi-equational theories and prove almost-linear Zarankiewicz bounds in every fixed arity $r\geq2$. If $T$ is a $1$-semi-equational theory, then, for every formula $φ$ and fixed $t,r\geq2$, there is a constant $c$ such that each $K_{t,\ldots,t}$-fre