The horizontal-section diameter conjecture for subgroups of product groups

Let $G\leq\operatorname{Sym}(n)$ be a transitive permutation subgroup. Let $G_{1},G_{2},\ldots,G_{k}$ be finite groups lying on a horizontal section of the tree built from $G$, and let $G'_{i}\leq G_{i}$ for each $1\leq i\leq k$. For a subgroup $H\leq G_{1}\times\cdots\times G_{k}$, define

$$H'=H\cap(G'_{1}\times\cdots\times G'_{k}).$$

Horizontal-section diameter conjecture. There are absolute constants $C_{1},C_{2}>0$ such that

$$\operatorname{diam}(H)\leq C_{1}k^{C_{2}}\cdot\max\{[G_{i}:G'_{i}]\mid 1\leq i\leq k\}\cdot\operatorname{diam}(H').$$

This conjecture identifies the gap in the argument for small indices: it would control the diameter of $H$ in terms of the subgroup $H'$ and the coordinate indices by a polynomial in the number of factors. The source presents it as the needed conjectural ingredient, and no resolution is supplied here.