Not sure if they'd need a filing if they exceed 5% [of the convertibles] though.
No 13D filing necessary, since Sec. 13d applies only to equity securities:
"Reports by persons acquiring more than five per centum of certain classes of securities
Any person who, after acquiring directly or indirectly the beneficial ownership of any equity security of a class which is registered pursuant to section 12 . . . is directly or indirectly the beneficial owner of more than 5 per centum of such class shall, within ten days after such acquisition, send to the issuer of the security at its principal executive office, by registered or certified mail, send to each exchange where the security is traded, and filed with the Commission, a [Form 13D]."
law.uc.edu
Here's SEC rule 13d-1(a):
"Any person who, after acquiring directly or indirectly the beneficial ownership of any equity security of a class which is specified in paragraph (i) of this section, is directly or indirectly the beneficial owner of more than five percent of the class shall, within 10 days after the acquisition, file with the Commission, a statement containing the information required by Schedule 13D."
law.uc.edu
As for Jonathan's reasonable caution about the "creeping tender offer" rules, as I recall these are judicial glosses on Section 14d and the SEC rules under it, necessary to carry out the intent of those statutory provisions or some such reasoning.
But Sec. 14d and the rules under it seem also to be confined to "equity securities". Accordingly, it would be a pretty long stretch to bring the quiet accumulation of far-out-of-the-money convertible securities under that doctrine. (Not beyond what some lawyers might argue, but unlikely to be a winner, IMO.)
Sec. 14d: law.uc.edu |