Actually, "compulsory execution" is a legal term meaning that, having proved the right of yor claim in front of a judge (in this case there was no jugdment as everyone saw the crime), the state has the obligation to retribute you by taking any necessary measure against the lossing party (like seize a property or sell any good to pay the debt, also called execution of a property, hence the name).
So, in this case, it's not a death penalty, but he has to retribute the 90 gold coins that the officer claims that Satou's potions were priced, so, if he can't pay the money, the state is obligated to find a way to pay the debt (asuming that he doesn't have the money, as he already lost his former job, selling him to slavers might be the only way the state has to force him to repay his debt in this world).
So, he is alive. The other noble's fate is less clear in this chapter, but there is no reason to believe he was forgiven his treason (as was shown it was considered a grave crime in their legal code).