The fabulist Alex Jones and his company Infowars face a second and potentially ruinous trial beginning on Tuesday to determine how much they must pay the families of eight Sandy Hook shooting victims and an F.B.I. agent who responded to the attack.

Mr. Jones for years promoted lies on his radio and online show asserting that the 2012 Sandy Hook Elementary School shooting was staged by the federal government as a pretext for gun control, subjecting the families to a barrage of accusations and threats. This case is one of several he lost by default late last year.

The trial will take place in Connecticut Superior Court in Waterbury, less than 20 miles from Newtown, where the shooting occurred. Just hours after the massacre, Mr. Jones cast Sandy Hook victims’ relatives as complicit in his imagined government plot, in effect accusing them of faking the massacre that killed 20 first graders and six educators on Dec. 14, 2012. People who believe Mr. Jones’s bogus claims have tormented the families online, confronted them on the street, defaced and stolen memorials to their murdered loved ones, and threatened their lives.

In mid-2018, the families of 10 Sandy Hook victims filed four separate defamation cases against Mr. Jones. He lost all four lawsuits by default late last year, because he repeatedly refused to provide documents and testimony ordered by the courts. That set in motion three trials for damages, in which juries will decide how much Mr. Jones and Infowars must pay the families in compensatory and punitive damages. This trial is the second of the three.

In the first trial, earlier this summer in Austin, Texas, a jury awarded Scarlett Lewis and Neil Heslin, whose son Jesse Lewis died at Sandy Hook, nearly $50 million in compensatory and punitive damages, though Texas law caps the $45 million punitive award at $750,000 for each parent.

But the Connecticut case could prove far more costly to Mr. Jones. He has earned more than $50 million in annual revenue in recent years by selling diet supplements, survivalist gear and other merchandise on his conspiracy-laden broadcasts. The families in the Connecticut case sued Mr. Jones under the Connecticut Unfair Trade Practices Act, which does not limit punitive damage awards. If the Connecticut jury awards an amount similar to the damages in Texas or more, the families could collect it.

That prospect has prompted Mr. Jones to repeatedly try to delay Tuesday’s courtroom reckoning, including by putting his business into bankruptcy, twice. Last month, the Sandy Hook families filed a motion in the case accusing Mr. Jones of hiding assets and improperly channeling money to himself and his family, while claiming he is broke. They asked the bankruptcy court to order Mr. Jones to cede control over the finances of the parent company to Infowars, Free Speech Systems, to a trustee already monitoring the case.

Mr. Jones, a vocal ally of former President Donald J. Trump, is also under scrutiny for his role in events surrounding the Jan. 6 attack on the Capitol. He has also used the Sandy Hook lawsuits to pursue product sales and donations, while claiming the lawsuits are an effort by his political enemies to destroy him.