Some commenters on this post have suggested that this is simply not possible as a matter of text. But it seems to me that one can think of other parallel sentences where it is indeed permissible, given the context and common sense, to think that the “during” clause modifies both sets of verbs.
The soldiers were authorized to shoot any Germans they encountered in the trenches during the war.
It seems permissible to read this to be limited to those who are both shot and encountered during the war — you can’t shoot somebody 20 years later just because you saw them in the trenches back then.
The trial judge may exclude from the courtroom any spectator who is disruptive during the trial.
Again, it seems permissible to read “during the trial” to modify both “exclude” and “is disruptive.” The judge couldn’t necessarily exclude people who’d been disruptive some other time, and he couldn’t necessarily exclude people permanently from that room once the trial was over.
Don’t say “hi” to anybody you see at the racetrack.
One can certainly think of other constructions that might not be read this way, but I hope these examples show that this kind of reading is at a minimum grammatically possible, if the purpose or context of the sentence gives you reason to think it is meant that way.