Weaker distance-tree embedding conjecture for finite vector spaces

Let $q$ be a prime power, let $d$ be a positive integer, and let $\Delta\geq 2$. A distance tree is a tree whose edges are assigned nonzero distances in $\mathbb{F}_q^d$, and $\Delta(\mathcal{T})$ denotes its maximum degree. Weaker distance-tree conjecture. There is a constant $C_\Delta$ such that every $S\subseteq\mathbb{F}_q^d$ with

$$|S|\geq C_\Delta^{o(d\log q)}q^{(d+1)/2}$$

contains every distance tree $\mathcal{T}$ with at most $|S|-100q^{(d+1)/2}$ vertices and

$$\Delta(\mathcal{T})\leq\Delta.$$

This weaker version replaces the conjectured error term $100q^{d+1}|S|^{-1}$ by the larger uniform error term $100q^{(d+1)/2}$, while retaining the near-spectral-size hypothesis on $S$.