Factorization-pattern conjecture for polynomials in short intervals

Let $A$ be the polynomial ring over a finite field of size $q$. For a monic polynomial $f\in A$ of degree $d$ and a positive integer $m<\deg(f)$, let

$$\mathcal{I}_m(f):=\{f+a\mid a\in A,\ \deg(a)\leq m\}.$$

For a partition $\lambda$ of $d$, let $\pi_q(\mathcal{I}_m(f),\lambda)$ be the number of polynomials in $\mathcal{I}_m(f)$ whose irreducible-factor degrees induce the partition $\lambda$, and let $P(\lambda)$ be the proportion of permutations in $S_d$ having cycle type $\lambda$. Factorization-pattern conjecture. For all monic $f\in A$ of degree $d$ such that $3<d<\sqrt{q}/2$ and for all partitions $\lambda$ of $d$,

$$\pi_q(\mathcal{I}_m(f),\lambda)\sim P(\lambda)|\mathcal{I}_m(f)|\qquad\text{as }q^d\to\infty.$$

This conjecture asserts that factorization patterns of polynomials in sufficiently short intervals are distributed according to the cycle-type distribution of permutations. The cited work of Bank, Bary-Soroker and Rosenzweig proves the analogous asymptotic when $q\to\infty$ with $d$ fixed, while the stated uniform regime remains conjectural.