This was probably just before Mai carried out her plan to run away from home.
Literal
Mai [subject-が] running-away [object-を] carry-out, just-before thing [copula-だろう].
敢行する (かんこうする, 'to carry out boldly, to dare to execute') elevates a child running away from home to the level of a military operation — probably intentional dramatic irony. 家出 (いえで, 'running away from home') is the specific term. 直前のことだろう uses ~だろう for inference ('it was probably'). The sentence is narration looking back at a sequence of events.