Short answer. No. Article 1491 of the Civil Code says the listed persons cannot acquire the property by purchase even at a public or judicial auction, whether in person or through another. The public nature of an auction does not lift a prohibition tied to the buyer's position of trust over that property.
What the law says
cannot acquire by purchase, even at a public or judicial auction, either in person or through the mediation of another
Civil Code, Article 1491 — Persons Who Cannot Buy. Read the full provision →
The ban expressly reaches auctions
The tempting workaround is to say the sale was open and competitive, so surely a barred person can bid like anyone else. The article forecloses exactly that. The listed persons cannot acquire by purchase, even at a public or judicial auction, either in person or through the mediation of another. The words even at a public or judicial auction are there precisely because someone would otherwise argue the open format cleanses the purchase. It does not. A person who falls within the prohibition is barred from buying that particular property whether the sale is a private one or a public or court-conducted auction.
Who the prohibition covers
The article targets people whose position gives them influence or duty over the very property in question. It reaches, among others, a guardian as to property under his guardianship, an agent as to property entrusted to him, executors and administrators as to the estate they administer, and certain public officers, judges, and lawyers as to property in litigation or otherwise connected with their functions. The thread running through the list is a conflict between the person's duty to protect or deal fairly with the property and his private interest in acquiring it for himself. The auction rule stops that conflict being laundered through a public sale.
Why the format cannot save the sale
The prohibition exists to prevent people in positions of trust from turning that trust to their own advantage, and an auction does not remove the danger it guards against. A guardian or administrator still knows the property intimately, may influence how and when it is sold, and remains torn between the interest he is supposed to protect and his own desire to own it. Allowing him to buy simply because bidding was public would defeat the rule entirely. So the law treats an acquisition through auction the same as any other purchase by a disqualified person: it is not permitted.
The takeaway for a disqualified buyer
If you hold one of these positions of trust over property, do not assume you may bid for it at a foreclosure, execution, or estate auction. The safer course is to stay out of the sale of property you are charged with, or to remove yourself from the conflict entirely before any acquisition is contemplated, and, where the situation allows for it, to obtain the property owner's or principal's proper consent in the cases the law permits. A purchase made in defiance of this prohibition is exposed to serious challenge, and the openness of the auction will not be the shield a buyer hopes it to be.