In the past year, consumer giant Johnson & Johnson has been hit with four jury verdicts totaling about $300 million in cases linking its popular baby powder to ovarian cancer. With each defeat, its lawyers have pledged the fight is not over.

On Wednesday, Johnson & Johnson got its first crack at dismantling one of the verdicts in which Missouri juries have found that its baby powder caused a woman to get ovarian cancer. And a lot rides on how the U.S. Supreme Court rules in Bristol-Myers Squibb v. Superior Court of California, a closely watched case that could clarify the rules of jurisdiction for mass torts claims.

In oral arguments, broadcast live by Courtroom View Network, Johnson & Johnson attorney Thomas Weaver asked the Missouri Court of Appeals to reverse a $72 million award to the family of an Alabama woman who died from ovarian cancer. The panel's decision is likely to impact the three other verdicts in Missouri that went against Johnson & Johnson, including a $110 million verdict this month. (The company so far has prevailed in one trial.)