Keeps mum close, and slumps, oddly (5,2)

I believe the answer is:

shuts up

'keeps mum' is the definition.
('shut up' can be a synonym of 'keep mum')

'close and slumps oddly' is the wordplay.
'close' becomes 'shut' ('shut' can be a synonym of 'close'**).
'and' is a charade indicator (letters next to each other).
'oddly' indicates alternate letters.
The alternating letters of 'slumps' are 'sup'.
'shut'+'sup'='SHUTS UP'

I've seen this clue in The Telegraph.

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