The correct option is A As
When a sentence/clause is dependent on another sentence/clause, then the dependent clause ("the lawyer contacted his son with news of his inheritance.") is a subordinate. When these two are combined, a subordinating conjunction is used. The events in the main clause (Mr. Harrison passed away) and subordinate clause (lawyer contacted his son) are cause and effect, when Mr. Harrison passed away, only then did the lawyer call his son, hence the conjunction 'as' is used, 'as' denotes 'the result', and option B is thus, correct. The underlined conjunction 'though' is incorrect because it's used when the sentences are contrast in meaning, and the sentences here do not have opposite meanings. Hence option A is incorrect. 'If' is a conjunction used for conditional sentences, hypothetical situations which may or may not come to being. As none of the given sentences are hypothetical, the events did take place. When a sentence begins with 'if' there's a would at the second part of the sentence, because both the events would be hypothetical, hence 'if' will not be used, option C is incorrect. Similarly, 'should' is a conjunction used for conditional sentences, hence option D is also incorrect.