Learn Spanish (Latin America): Aun prolongando el diálogo, no acabaron por mediar en la disputa. - Even prolonging the dialogue, they did not end up mediating in the dispute.

Yes! Both acabar por and terminar por followed by an infinitive mean 'to end up doing' something. In this sentence, no acabaron por mediar means 'they did not end up mediating.' These are culminating periphrases, meaning they highlight the final outcome of a long process. Here, the process was the prolonged dialogue that ultimately failed to result in a mediation.

Why is Aun spelled without an accent mark in this sentence?

Without the written accent, aun means 'even' (often used to show concession or contrast). If it had an accent (aún), it would mean 'still' or 'yet'. Here, it means 'even,' setting up the contrast: even prolonging the dialogue, they didn't mediate.

Is it common to use a gerund like prolongando right after aun?

Yes, this is a common and very natural structure in Spanish. The combination of aun plus a gerund, like aun prolongando, translates to 'even doing' something or 'even while doing' something. It introduces a concession, showing that despite the ongoing action of the dialogue being prolonged, the expected result didn't happen.

How does no acabaron por mediar feel different from just saying no mediaron?