You’re getting too tied up in the legal weeds. If it was actually immaterial, they wouldn’t have lied in sworn statements to courts.
The purpose of the lie isn’t to meet a legal element of the claim. It’s rhetorical. It’s to give the right wing legal movement, and the justices themselves, “facts” that they can bolster the persuasiveness of their arguments with. A boring deck action doesn’t have the same public impact as “heathen f**s tried to force this good Christian woman to endorse their sinner lifestyle! Radical left embraces satan once again!”
The Republican justices have a symbiotic relationship with the right wing legal movement that brings bullshit like this. The lie helps them justify a decision like this to the public and entertaining it rewards the activists for bringing the types of cases they want them to bring.
We know how loose they’ve been willing to be with the facts in cases in the past, but why wouldn’t they knowingly entertain wholesale fabrications? Even Roberts is happy to blatantly twist the facts when it’s clear that he’s doing so. If they think the public won’t find out certain claims fundamental to the public presentation of a case are complete bullshit, there’s no reason at all to think they wouldn’t be happy to go along. I’d bet even a pretty cursory investigation of other cases brought by the ADF and similar groups would reveal similar fabrications.