Bain Capital is increasing its offer for Fuji Soft with a bid that values the Japanese software company at $4.3 billion, extending a rare battle with KKR in the country’s booming buyout market.

The Boston-based investment firm plans to pay ¥9,600 a share for Fuji Soft, it said in a statement on Wednesday. That’s 1.6% higher than KKR’s offer of ¥9,451 and values the company at around ¥647 billion ($4.3 billion).