Simple lifted caps avoid the original essential loop and a fixed intrinsic neighbourhood

Use the one global positive smooth field $V$ fixed before cap development in The canonical Jordan cap bundle develops coherently over every positive band, agreeing with the original fence field near its compact trace. Thus the whole compact leaf $A$, when it is compact, and every intrinsic compact set used below lie in $\mathrm{dom}V$. If the reduced original leaf $A$ is compact, shrink its positive fence using compact separation for $K=A$ by [F1], so that its positive boundaries and entire cap leaves are distinct from $A$; every cap then misses the entire $A$, and uniform neighbourhood avoidance is automatic.

If $A$ is noncompact, its fundamental group is torsion-free by the finite surface adapter of [F1]. Short fixed-flow separation for the compact set $K=\gamma (S^1)$ makes every boundary ${\gamma }_t$ disjoint from $\gamma$. If a projected disk cap met $\gamma$, its leaf would be the original leaf $L$; lift the crossing to the disk in the universal cover of $L$. The lift of $\gamma$ through that crossing stays inside the disk, because it cannot cross the disk boundary (its projection is disjoint from ${\gamma }_t$); the next lift of $\gamma$, starting at the endpoint of the first, also stays inside the disk, and inductively the whole orbit under the nonidentity deck element $\alpha$ represented by the essential loop $\gamma$ lies in the compact lifted disk. All these orbit points lie in one covering fiber, whose intersection with the compact lifted disk is finite by F2, so $\alpha$ has finite order, contradicting torsion-freeness of the oriented surface group. Hence the projected cap misses $\gamma$.

Choose a relatively compact intrinsic neighbourhood $S$ of $\gamma$ in $L$ and a smaller connected-near-$\gamma$ neighbourhood $S^{\mathrm{\prime }}$ with closure contained in $S$, so that every point of $S^{\mathrm{\prime }}$ can be joined to $\gamma$ by a path in $S$ (finitely many leaf charts suffice). Compact flow separation for $\bar{S}$, not merely for $\gamma$, ensures every short ${\gamma }_t$ avoids $S$. If a projected cap on $L$ met $S^{\mathrm{\prime }}$, lift a path in $S$ from that point to $\gamma$ starting inside the lifted disk; it cannot leave the disk, because crossing its boundary would project to an intersection of ${\gamma }_t$ with $S$, so its endpoint lies inside the disk and projects to $\gamma$, contradicting step 2.1. Thus all these cap images avoid $S^{\mathrm{\prime }}$ uniformly, and caps on other leaves are automatically disjoint from $S^{\mathrm{\prime }}$. This corrects the source's unsupported uniform intrinsic-distance assertion by applying compact separation to an enlarged intrinsic compact neighbourhood; the argument uses finitely many charts and the one fence, hence only the standing countable choice from [F3].