Does difficulty adjustment exploiting selfish mining assume pool loyalty?

The 2018 paper "On profitability of selfish mining" by Cyril Grunspan and Ricardo Pérez-Marco [0] introduces a selfish-mining attack that relies on the difficulty adjustment algorithm. Their followup paper (same year) "On profitability of stubborn mining" [2] extrapolates this to attack variants.

It first explains how, when difficulty is constant, selfish-mining is less profitable than honest mining (see also [1]). But then it claims that the difficulty adjustment algorithm introduces a way to make profit. Before I even attempt to understand the math, I'm having a hard time making sense of this intuitively.

By temporarily withholding blocks in period N the attacker is increasing the stale block rate, both for themselves and for honest miners. This causes a drop in difficulty which makes mining in the next period more profitable. So far I follow.

But why would a lower difficulty in period N + 1 benefit the selfish pool more than its competitors? It seems to me all pools profit equally in period N + 1. But the selfish pool had to make an investment in period N, since we've already established that at constant difficulty this strategy is less profitable.

So as a neutral miner choosing between pools, you don't want to be in the attacker pool in period N when it's losing money. You also don't want to join it in period N + 1 because you can reap the same reward in any other pool.

So are they making an assumption about pool loyalty?

[0] https://arxiv.org/abs/1805.08281

[1] https://bitcoin.stackexchange.com/a/125844/4948

[2] https://arxiv.org/abs/1808.01041



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