The Catholic Encyclopedia defines Canonical Admonitions, Volume I, 1907.
Here we see how seriously the Church regards suspicion, even based on mere public rumour, and what the Church's approach to such matters was historically.
“A preliminary means used by the Church towards a suspected person, as a preventive of harm or a remedy of evil. In the Instruction emanating in 1880, by direction of Leo XIII, from the Congregation of Bishops and Regulars to the bishops of Italy, and giving them the privilege to use a summary procedure in trials of the clergy for criminal or disciplinary transgressions, Article IV decrees: "Among the preservative measures are chiefly to be reckoned the spiritual retreat, admonitions, and injunctions"; Article VI: "The canonical admonitions may be made in a paternal and private manner (even by letter or by an intermediary person), or in legal form, but always in such a way that proof of their having been made shall remain on record."
“These admonitions are to be founded upon a suspicion of guilt excited by public rumour, and after an investigation to be made by one having due authority, with the result of establishing a reasonable basis for the suspicion. Upon slender foundation the superior should not even admonish, unless the suspected person has given on previous occasions serious motive for faultfinding. Admonitions may be either paternal or legal (canonical), If the grounds are such as to produce a serious likelihood, or half-proof, they will suffice for a paternal admonition, which is administered after the following manner: The prelate either personally or through a confidential delegate informs the suspected person of what has been said about him, without mentioning the source of information, and without threat, but urges amendment. If the party suspected can at once show that there is no basis for suspicion, nothing further is to be done in the matter. If his denial does not banish the doubts about him, the prelate should try by persuasion, exhortation, and beseechings to induce him to avoid whatever may be a near occasion of wrong, and to repair the harm or scandal given. If this is not effective, the prelate may begin the judicial procedure. If the proofs at hand are inadequate, this is not advisable; he should rather be content with watchfulness, and with using negative penalties, such as withholding special offices and, where no slur could be manifest on the suspect's reputation, by withdrawing those before held. If the suspect does not answer to the summons, the prelate's suspicion reasonably increases, and he should then depute a reliable person to seek an interview with him, and to report to him the result. If he should refuse to deal with the delegate, the latter in the name of the delegating prelate should through another or by letter send a second and a third peremptory call, and give proof of the further refusal, with evidence that the summons has been received; now the suspect is presumed guilty. Thus the way is paved for the above-mentioned canonical or legal admonition."