Why is the BBC saying Boris’s Savile claims are false?

Is Boris Johnson’s claim that Keir Starmer failed to prosecute Jimmy Savile ‘false’? The BBC certainly thinks so. During the Radio 6 bulletins last night, a BBC newsreader stated: 

The Commons Speaker has rebuked Boris Johnson over his false claim that Sir Keir Starmer failed to prosecute Jimmy Savile after the Labour leader was targetted by protesters. 

Last week, the Prime Minister said during a Commons statement that Starmer had spent his time as head of the Crown Prosecution Service ‘prosecuting journalists and failing to prosecute Jimmy Savile’....