The negotiations are unsuccessful, and the Boardsʼ hopes to continuing talks on a future deal seems to be increasingly improbable.

The underlined portion is an independent clause, and there are errors in it due to the following reasons. 

The subject in this case is "The boards' hopes", which is a plural subject (hopes refers to many hopes, i.e., plural), so the verb should agree with it. So the verb "seems" should actually be "seem" . Additionally, there is a mix-up between the infinitive and the gerund here. After "hopes to," standard grammar requires either an infinitive ("to continue") or a gerund ("of continuing").
The original uses "to continuing," which mixes the preposition "to" with a gerund, creating an ungrammatical construction.

Based on context, either "boards' " or "board's " is correct because the possessive form depends on the number of boards. So either of them is grammatically correct here. Even if there is "one" board, the "hopes" are plural, so the verb in either case should be "seem " and not seems".

Considering all this, we see that only Option A properly addresses both the subject-verb agreement issue and the infinitive/gerund problem, making it the only grammatically correct choice. The other options each violate at least one critical grammatical rule.Hence, Option A is the correct choice.