The correct option is A doubt
The given sentence consists of two clauses, "I...they'll accept the offer" and "it's worth a try", both are independent clauses as they're complete in meaning. The conjunction that combines these clauses is a coordinating conjunction "but" which is used when the clauses are contrast in meaning. Hence the verb that fits the sentence should be suitable to express a contrast meaning. From the options, 'suspect' and 'doubt' are often confused for each other, but they're opposite in meaning, 'suspect' means when you believe that someone is guilty, whereas 'doubt' means that you don't believe someone is innocent. Here, 'doubt' means that speaker does not believe that "they'll accept the offer" but in spite of that believes it's worth a try, whereas 'suspect' means the speaker believes they 'won't' accept the offer. However, the sentence is suited for 'doubt', it does not contain the negative 'won't'. Hence option C is correct and A is incorrect. 'Hope' means 'to feel something (that is desirable) will happen' and 'expect' is 'to anticipate the occurrence of something'. When each is placed in the sentence, it makes sense, but the second clause does not comply, because they're supposed to have contrast meanings, whereas 'hope' and 'expect' are both in the affirmative, and "worth a try" also connotes a positive meaning. Hence options B and D are incorrect.