The approach suggests that when a transaction is evicted due to not being included in a block, subsequent transactions from the same sender can be made to pay an increased minimum fee. This incremental rate adjustment is seen as a way to compensate for the initial failure to include the transaction, albeit in a limited and time-bound manner. This mechanism draws a parallel with the concept of allowing free relay for transactions that have remained in the mempool for an extended period, specifically those timing out after two weeks.

The notion of increasing fees for transactions following an eviction introduces a nuanced method to manage network resources and prioritize transactions based on their willingness to pay higher fees post-eviction. This strategy, however, is recognized as being less effective than directly harvesting fees from transactions. The conversation also touches upon the difficulty of implementing a "free relay at the bottom of the mempool" strategy, which becomes less feasible due to the proposed fee increment system. Based on insights from AJ, it's estimated that the typical duration for this decay in prioritization could range between 10 to 100 blocks, indicating a significant departure from the two-week grace period for free relay, yet offering a flexible framework for managing mempool congestion and transaction prioritization.