What he is, is unemployed. Pure and simple. Until the franchise tender is signed or withdrawn, Asante's an unemployed laborer in a unionized workforce.
Everybody has the right not to work if one can afford to do so.
Now, given his apparent love of money and the lifestyle it affords him, there's no way he's going to forfeit over half of his salary for potentially two years in a row.
Alonzo Shavers appears to be a savvy agent, with a good head for proper negotiation. It is much more likely that Asante cashes in on free agency if he reports to camp and is starting a cornerback week one. If he appears to pick up from where he left off in Indianapolis, Belichick may well offer him an extension during the bye week. I think it unlikely he will franchise him again. The other option is staying away until week 10, without the benefit of practicing and playing in games, and hoping he can show the league that he's worth a mega-deal. Which will be hard to do if he can't win his job back. It's entirely possible that he doesn't get onto the field until week 12 and plays nickelback for the rest of the season.
Bottom line: Asante reports before the end of training camp.