At the risk of being pedantic, as a Hawaiian frequent flyer I'm a bit offended that the OP asks whether they will honor elite status with another airline to give him free upgrades.
The verb honor implies an agreement that they have made. But they have not entered into any such agreement to honor someone who is a frequent flyer of another airline. And I'd be really annoyed if they did, as they don't even give those of us who are their own frequent flyers free upgrades!
For those of us who live outside the US (and I do, even though I fly around 30 domestic flights in the USA each year) we find "free" upgrades for elites one of the most pernicious aspects of the American aviation market. Because most of First Class is given away the standards are pathetic by global standards - somewhere between what we call Premium Economy and Business class - and the airlines see no reason to improve Economy class to an internationally accepted level because their elites just aspire to be gifted First.
The best thing about Hawaiian is that its standards are pitched part-way between US domestic standards and internationally accepted ones. Food is still free. IFE is being rolled out in coach. The trade-off is that if you want to sit in First Class you have to buy it. I like it!