That question was recently addressed by the Ohio Court of Appeals. The Court found that:
While [the law] affords trial courts discretion in imposing restrictions corresponding to a CPO, this discretion is not limitless. Restrictions must bear a sufficient nexus to the conduct that the trial court is attempting to prevent. This court has applied the requirement that the restriction on the right to bear arms be related to the conduct … [when] the testimony was that the respondent had threatened the petitioners that he would kill them while holding a gun and specifically stated that “I have a gun. I can kill you.” Since the conduct specifically was related to firearms, this court found a sufficient nexus between the restriction on access to firearms and the conduct to be prevented. ..
However, in cases where there was no sufficient nexus between the conduct and the firearm restriction, the restriction has not been permitted. …
Here, no testimony was presented that [the target of the CPO] even owned a firearm. No testimony was presented that he had ever threatened [the protected party] with a deadly weapon. According to the trial court, the only reason that the restriction was present was because it was on the preprinted form. Without a nexus between the offending conduct and the restriction, Stone’s constitutional right to bear arms may not be restricted….
(Quotation marks and citations omitted.)
At the Volokh Conspiracy, Eugene Volokh provides a longer excerpt from the case and notes that, “Not all courts follow this approach, but … some do.”