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It sounds like this is a complicated situation. Regarding the disavow, each file you upload completely replaces the previous one. So, if you had unnatural links and then you uploaded a new file that did not contain some of those unnatural links, you've essentially asked Google to start counting them again. It can take time for them to start to count again as Google has built a lag in to the system to prevent people from trying to game it. However, it's certainly possible that the unnatural links are hurting you. The main algorithm that uses data from unnatural links is Penguin and Penguin has not refreshed since 2014, so it is unlikely that Penguin is affecting the site. But, there are other algorithms that use links. If one of those is affecting you, then reinstating a full disavow file should show some positive effect within a couple of weeks. With that said, as far as we know, the Quality update was not related to backlink quality, but rather on-page quality. So, if you dropped because of the Quality update then there are likely other issues with the site.