It may be worth mentioning that you should not stop your research if you find out that the wallet is open source, because open source doesn't mean that the wallet is automatically secure, but it is surely a positive point to use a wallet. There are wallets that lie in their websites that they are open source and even have Github repositories that haven't been updated for a long time,
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Open Source is just a sign that the wallet can be developed by anyone and the code can be verified by the community.
But there is no guarantee that Open Source wallets will remain secure.
When the developer of an Open Source Wallet or a new wallet does not get much attention, there are no revisions or fixes for existing bugs, and no feature updates that must keep up with transaction developments.
Therefore, it is necessary to choose an Open Source wallet with a strong community so that updates can continue to be made.
I don't think it's being neglected on purpose, it's just that not many people have the skillset to do it properly or fully. I don't know how to do roofing or plumbing, so I can't bring anything useful to the table. It's the same with verifying the security of a piece of code.
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As @pmalek said, code testing is not neglected but it is not easy to improve or develop open source wallets without having expertise in wallet programming and so on.
This will be a task for developers who are already experts in the field.
We may only be able to contribute by using the wallet and reporting when some bugs appear.