The core issue at hand is whether the gathering theorem can still be applied when the optimization strategy involves moving a sublinearization of $L$ to the forefront of $L_{opt}$, rather than reordering sublinearizations within $L_{opt}$ itself. This distinction is crucial because it questions the applicability of the theorem under conditions where the optimization does not strictly adhere to rearranging elements within the original set but instead focuses on optimizing by repositioning elements from a related, yet distinct, set.

The gathering theorem, as mentioned, plays a pivotal role when dealing with single $L$-chunk scenarios. It asserts that if a payload ($p$) consists solely of an $L$-chunk, then aligning this chunk as per the theorem's guidance will yield the most efficient outcome—matching, and effectively maximizing, the feerate observed in the first chunk of the optimized list ($L_{opt}$). This premise hinges on the assumption that the optimization process is straightforward and directly applicable, leveraging the inherent structure of $L$-chunks for optimal rearrangement.

The crux of the problem lies in evaluating the theorem's versatility or adaptability when faced with a more complex optimization task. Specifically, the challenge arises in determining the theorem's effectiveness when the optimization strategy entails advancing a sublinear part of $L$, implying a form of pre-selection and reordering before applying the standard optimization logic traditionally envisioned for $L_{opt}$. This nuanced application scenario raises important questions about the boundaries and flexibility of the gathering theorem in handling cases that deviate from its conventional use case, thereby inviting a deeper examination of its foundational principles and potential limitations in accommodating variations in optimization strategies.