This is a bit similar to the "sidechain" idea I proposed here.

I think there's the same challenge to solve than in my proposal: The "miners" would have to be rewarded in some way to incentive them to store the correct contents. But how can you be sure that they store the correct content, and not some randomly generated fake posts which would take them probably less time than to scrap the forum?

Perhaps this could be solved in the same way like Optimistic Rollups work. The rewards to the "forum miners" are locked for some time, let's say a week or two. If there is somebody who discovers that the contents added by a "miner" don't exist or differ from the forum contents, they can post a proof. The reward is then slashed and the miner perhaps even penalized in another way (slashing a security deposit, for example), and the fake content is pruned, and an additional "miner" gets the opportunity to store the missing 10-minute window.

Of course this raises additional challenges: there may be miners trying to attack the forum and store the randomly generated fake posts there instead of the "real" posts, to be able to get the reward. Perhaps as an option of last resort a DAO vote (for example via PoS) could he held if there's some doubt about the correct forum content. Just to ensure there's not the slightest incentive to try such attacks.