Defendant is charged with murdering his alleged mistress, Ms. Stewart. The State's theory was that Defendant, from the very inception of his marriage to Ms. Brown, was romantically involved with Ms. Stewart, that Ms. Stewart became pregnant as a result of the affair, that Defendant insisted that she abort the pregnancy, that she refused, and that he killed her because he feared that the pregnancy would wreck his marriage. Defendant made clear, both at the outset and throughout the trial, that his defense was based on the proposition that his wife, who was aware of his affair with the victim and had threatened both him and the victim in the past, killed the victim out of jealousy. In its case-in-chief, the State called Ms. Brown, who recounted that she and Defendant were married on August 20, 2014, and that on September 9, 2014—the day they signed a lease on their new apartment—she discovered a picture of Ms. Stewart in Defendant's car and thus learned that he had a girlfriend. The problem was exacerbated by the fact that Ms. Stewart continued to callDefendant, which led to arguments between him and Ms. Brown. Ms. Brown had a number of conversations with Ms. Stewart, complaining about her calls to Defendant. They were all "heated discussions," she said. At one point, when Defendant said that he had a doctor's appointment and would be late, Ms. Brown discovered him and Ms. Stewart together at a bowling alley, which led to another argument and to Ms. Brown throwing a bottle at the victim. All of this testimony was admitted without objection. Ms. Brown then testified that on September 9, 2015—the night of the murder—Defendant returned home at around 4:00 a.m., that she asked him where he had been and that he refused to tell her. Ms. Brown then got into an argument with Defendant about his talking with the victim. In response to the question, "What happened then," Ms. Brown said, apparently to everyone's surprise, "He told me he killed her and I didn't believe him." Defendant objected, asserting the inter-spousal communications privilege.
Had the Defendant objected to Ms. Brown's testimony regarding the events leading up to the night of the murder, would should they have been admitted? Why or why not?