There is another legal system where participants face a similar threat, where participants believe that false testimony has dire consequences even if no witness could otherwise prove they're lying. So we can look to that system for guidance on how this one might play out.
I'm talking about rabbinic courts, specifically in the time of the Sanhedrin. The ultimate "stick" that the court had was to require somebody to swear an oath. What's the big deal about swearing an oath? To modern-day Americans, maybe not much -- just words and all that. But in its original context, a false oath was (and to many today still is) a transgression against God. Who, you know, would punish you for that. So you really, really don't want to swear a false oath, and it's safer to avoid swearing an oath at all because of a reluctance to bring out the "big guns", so to speak. The passages in the talmud that talk about how the Sanhedrin operated and what the laws of various cases are show a tendency to avoid oaths at all unless necessary. The truth stone, being not only a big gun but a singleton, will be similarly reserved for special cases. Most trials will be conducted as they were before.
Second, when an oath was required, its text was fixed. I'm not sure if it's fixed by the law itself or by the judges, but either way, the accused (or witness) doesn't get to say "sure I'll swear, and what I swear is (carefully-crafted dodge)". When your truth stone is brought into a case, expect the one presiding over the case to similarly dictate what the speaker must testify to.
Just having the "truth" isn't enough to settle some cases, though. Your truth stone, and an oath, measures the speaker's perspective. It doesn't measure actual truth. You could therefore end up with situations where two litigants both have the "truth" on their side, but their positions are incompatible. One of them is mistaken, but not lying. ("I picked up the lost bag of gold first!" "No, I did!" They both ended up holding onto it; who was really first? You don't know.) You will have cases where the truth stone didn't help you, in other words, and you should expect to fall back on what you would have done without it. (Sometimes rabbinic cases end with no oaths and "they divide the proceeds".)
A final note: the rabbinic system I'm describing didn't rely only on the fear of false oaths, any more than the king here should rely only on the truth stone. Testimony of eye-witnesses and evidence are still big factors, same as with courts today. Using the truth stone is still the exception, as I argued earlier.