Yes, your wallet has to be kept open 24 hours a day. This is really done in such a way that, if you secure your wallet, your Gulden are 100% safe because your wallet will be encrypted all the time and only you can access it.
Of course a desktop uses a lot of power, a laptop less but still it is not very desirable to leave your laptop on all the time. Therefore a community member has developed a wallet for a Raspberry Pi microcomputer that uses only +/- 3 Watts. see G-Dash. There is also a Cloud version developed by the same community member.

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