Structural conjecture for near-extremal joints configurations

Let $\mathbb{F}$ be a field, and let $(\mathcal{J},\mathcal{L})$ be a joints configuration in $\mathbb{F}^d$, with $J=|\mathcal{J}|$ joints and $L=|\mathcal{L}|$ lines. Assume that, for some $x\geq d$, $J=(1-o(1))\binom{x}{d}$ and $L=\binom{x}{d-1}$. Let the structural property in Lemma Structural be the property referred to in the source. Structural conjecture. There exists a subset $\mathcal{J}'\subseteq\mathcal{J}$ with $|\mathcal{J}'|=(1-o(1))J$ satisfying that structural property with $\mathcal{L}$. The conjecture predicts that asymptotically optimal configurations retain the expected combinatorial structure after discarding a negligible proportion of joints; the supplied excerpt does not state the content of Lemma Structural, so that detail remains to be verified.