Place restrictions on what can be reported on in an inquest. Make it a requirement for the next of kin to consent to information shared.
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After sitting through an inquest surrounding the death of my dad then later that day realised it had been shared via local papers. This is hugely distressing for family members including young adults. The information shared was medical history, name, age and much more.
We have a great deal of data protection rights while we’re alive this should stand when deceased.