On this website's page:
... under the heading of "Upgrading your Battle Pets"...
There is no mention as to whether or not an acquired "Stone" is an account-wide item or an item specific to whichever char obtained it (Soulbound). I spent the last few hours trying to locate a (Flawless "Family-related') stone which I received via one of my (too many!) char's, and cannot find it. I still haven't gone through all of my char's, but I felt the need to look on this website first. No related search info on on your website via the keyword: "stone" or "stones."
I use the Carbonite addon and its "Warehouse" feature lists everything that my char's have, including what they have in their banks. But, for some reason, I can't find the stone I aquired. I don't remember the exact name of that stone, which makes things even more difficult. I do remember that it can update a Battle-pet to Rare, as long as its in the "Beast" family.
Is it an account-wide item (once received)?
I think a note on your website page about that would be helpful.