Towards a CFSG-free diameter bound for $\mathrm{Alt}(n)$

Helfgott and Seress have proved the existence of a quasipolynomial upper bound on the diameter of $\mathrm{Alt}(n)$. In this paper, we walk partway towards removing the dependence on CFSG from that result, by using the algorithm solving the string isomorphism problem (due to Babai) in its CFSG-free version (due to Babai and Pyber): the result contained in here relies on the analysis of Babai's algorithm contained in Dona, based in turn on Helfgott. Conditional on a conjecture about certain products of small-indexed subgroups (Conjecture 4.5), we provide a CFSG-free proof of a bound on the diameter of $\mathrm{Alt}(n)$ that is better than the already existing CFSG-free results in the literature. In fact, the same bound holds for all transitive permutation subgroups $G\leq\mathrm{Sym}(n)$. The paper is part of the author's doctoral thesis.