Why does each family think it should have the disputed land in "The Interlopers" by Saki?

Based upon the little information that provides "backstory" (what happened before the events of the story) in "The Interlopers," the debate argument Ulrich Gradwitz could best use to state a claim for why his family should have the land is that his family legally owns it and seems to have long owned it, because the Znaeym family at some date took possession of it illegally. The debate argument Georg Znaeym could best use to state a claim for why his family should have the land is that his family took possession of it away from the Gradwitzs.

[T]he narrow strip of precipitous woodland that lay on [the] outskirt [of the forest lands of Gradwitz] was not remarkable. . . but it was the most jealously guarded of all its owner's territorial possessions. A famous law suit, in the days of his grandfather, had wrested it from the illegal possession of a neighbouring family of petty landowners; the dispossessed party had never acquiesced in the judgment of the Courts.

It will be very difficult to support the Znaeym argument in a debate—they failed to support their argument in court—because the argument of I possess what is legally not mine to possess is a very weak argument, even if the Karpathian Common Law at that time upheld the idea that possession is nine-tenths of the law. In support of the difficulty of making this argument, the Karpathian "Courts" that heard Ulrich's grandfather's case against the Znaeyms three generations earlier did not rule in favor of the family that possessed the land (the Znaeyms). On the other hand, it is significantly easier to support the von Gradwitz argument in a debate because their claim is that legal ownership equates with present and future possession.