They are being ordered to disclose and they are refusing to do it. The article says the lawyer requested to be held in "friendly contempt." I'm pretty sure that doesn't exist. It would probably be more accurate to say that the lawyer conceded the company was in contempt (i.e. they were refusing the order to produce and maybe even conceding it was legal) and was requesting a light sanction. The strategy seems rather bold.

I am not versed in Illinois contempt law, but judges have very broad constitutional authority to sanction people or companies for intentional failure to comply with a lawful order. In this case, you could expect the judge to do something like order the company to pay five thousand dollars per day until they disclose the records.

There could be an argument that the judge could legally order the sheriff to seize their servers or throw the CEO in jail until the company complies, but for some reason stuff like that doesn't seem to ever happen in the corporate context.