Well the short answer to this question is that backlinks will be recognized as long as they exist. That doesn't mean they will always be of value. Everytime a crawler visits a website, it recognizes all the backlinks on each page and then the algorithms decide how to index them. They might be counted to help the linked page, they might be discounted, or they might be counted as a negative for the linked page. Google has changed how it deals with backlinks a lot over the years and will continue to do so.
The OP question though leads to some additional questions though. Each time the crawler sees a backlink, one of the first things the algorithms probably check is whether it's already been dealt with by the index.
So here's a question for anyone who has done some testing and study of the matter.
What do the algorithms do when it finds the backlinks in the index change relevancy. It obviously deals with the case of relevant backlinks that become irrelevant after a website changes to a different purpose. Actually I guess I'm more curious in what happens after google decides those backlinks are irrelevant and discounts them or even counts them as negative. What happens if a few years later an old domain with irrelevant backlinks is redesigned so that the backlinks become relevant again.
Say there was a bicycle info website with tons of relevant links. But the domain expired and repurchased for some kind of shallow spammy backlinking profile. But then later expires again and is redesigned again for the original purpose and pages made which are relevant to all the older backlinks. Will the algorithms start counting those older backlinks as relevant again as the crawlers come across them again? If so, will that possibly even trigger the algorithm to have crawlers recheck other older backlinks in the index that were discounted before?
The OP raises some interesting questions. Hopefully some of the people that do a lot of testing for these kind of things might have some good answers.