In Julius Caesar, what request do the conspirators make just before killing Caesar?
The conspirators in Act 3 fear that their conspiracy has been detected. So, they try to hasten the plot and get close to Caesar. The way they do this is by asking for Caesar's clemency on an exile. First, Metellus goes to Caesar and asks for the return of his brother, Publius Cimber, to Rome. Metellus asks:
"Is there no voice more worthy than my own To sound more sweetly in great Caesar's ear For the repealing of my banish'd brother?"
Caesar refuses, because Cimber has been exiled for lawful reasons. At this point, the same plea is reiterated, but this time also by Brutus and Cassius, and the scene is even more serious, as they both kneel as if to beg Caesar. Caesar is still unyielding. Brutus says:
"I kiss thy hand, but not in flattery, Caesar; Desiring thee that Publius Cimber may Have an immediate freedom of repeal."
"Pardon, Caesar; Caesar, pardon: As low as to thy foot doth Cassius fall, To beg enfranchisement for Publius Cimber."
At this point Cinna also kneels and asks for Caesar's favor in the matter. Caesar refuses again. To make the scene even more powerful and ominous, Decius and Ligairus also kneel. It is at this point that first blood is drawn by Casca.
At the beginning of Act III, Julius Caesar arrives at the Capitol. There Artemidorus pleads with Caesar to read the paper he hands him, and when Decius requests that Caesar read something else, Artemidorus interjects, "Delay not, Caesar; read it instantly" (3.1.6); unfortunately, the effect of his saying this is counter to Artemidorus's intention: Caesar asks if the man is mad and does not read it.
Having witnessed these exchanges, the conspirators become anxious that Caesar may be informed of danger; so, Cassius tells Casca to do something quickly. Decius suggests that they locate Metellus Cimber who has a petition requesting the repeal of a decree of banishment imposed by Caesar upon his brother, Publius Cimber. Hence, Metellus Cimber hurries to Caesar in order to make this request, followed by Brutus, who asks Caesar to consider repealing the banishment. Then, Cassius follows suit and even humbles himself before Caesar,
As low as to thy foot doth Cassius fall
To beg enfranchisement for Publius Cimber (3.1.55-56)
By the time Caesar replies, he is in position to be assassinated.
The answer to this question can be found in Act III scene 1 of this play. The conspirators gather to ask for clemency for the brother of Metellus. Note how Brutus presents the plea:
I kiss thy hand, but not in flattery, Caesar,
Desiring thee that Publius Cimber may
Have an immediate freedom of repeal.
Publius Cimber was banished by Caesar, as we are told by Metellus, and it is clear that the conspirators are using this as a test to demonstrate Caesar's despotism and the unyielding nature of his power. It is his response, stating how he is as "constant as the Northern Star," that triggers the ire of the conspirators and leads to Caesar's assassination. He shows himself to be inflexible and unyielding--precisely the qualities that Brutus and the other conspirators are so concerned about.