Fixed-point boundedness and the expressive power of infinitary logic

Let $K$ be a class of finite structures, let $t(n)$ be polynomially bounded, and let $r(s, \mathcal{A})$ denote the relevant iteration-depth parameter for a structure $\mathcal{A}$ and positive integer $s$. The class $K$ is $t(n)$-fixed-point bounded if, for every first-order formula $\varphi(X,\overline{x})$ positive in $X$ with free variables among $\overline{x},X$, there is a constant $c$ such that, for every positive integer $n$ and every structure $\mathcal{A}$ of size $n$ in $K$, the depth of $\varphi$ in $\mathcal{A}$ is at most $ct(n)$. Fixed-point boundedness conjecture. For every class $K$ of finite structures and every polynomially bounded $t(n)$, the following are equivalent: (i) $K$ is $t(n)$-fixed-point bounded; (ii) for every $s$, there is a constant $c$ such that, for every positive integer $n$ and every structure $\mathcal{A}$ of size $n$ in $K$,

$$r(s,\mathcal{A})\leq ct(n);$$

(iii) on $K$, every $\textrm{L}_{\infty\omega}^{\omega}$-formula is equivalent to an $\textrm{IND}[t(n)]$-formula. This conjecture seeks to characterize exactly when fixed-point iteration has polynomially bounded depth and when the corresponding infinitary and inductive logics have no greater expressive power on $K$.