My hubby was offline for over a month at least and when he logged back in, there were no rewards at all, even though he is on steemauto to upvote people. Although he is also new and only has an SP of about 10 right now. Can you earn curation rewards at all if your SP is that low?
An upvote with 10 sp will only have a 0.001 or less upvote value. Honestly, that isn't much. you will almost never get a curation reward from sp of that amount. and if you do get some it will not be much. And If you stay offline you will not get any rewards. Qnd this case your husband was offline for almost a month. Why would you? you are not curating or posting something that would allow you to gather any rewards at all.
Now the answer to you question is yes, They will just stay there for ever, unless steem pulls back the uncollected rewards by adding a protocolnin a future hardfork that will allow them to do so. But until then your rewards will stay there. Just explore Steem "FAQ" section and you'll find all your answers there.
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No, you don't lose the SBD or Steemit in your wallet if you don't claim them. The money is already in your wallet, and since you alone have the key to it (unless you decide to share it with others), it will be there for as long as you want.
I have seen a few inactive steemit accounts with some SBD or Steemit in their account. The money is left there until they decide to claim it.