Story attached sentiments
To hold and have a sentiment is to have a feeling about something.
For something to have sentimental value means we have strong feelings about it.
Strong feelings have rich and deep stories attached to it.
What then about the objects that we say have sentimental value, but have no story attached to it? Those things we were handed with great gravitas? We have none of our own feelings about those.
Only keep the sentiments with stories attached.