Replace spaces with underscore after some pattern in line

How to replace all of the spaces in each line after XXX string?

Input sample data:

aaaaaa bbbbbb cccc XXX aaaaa bbbbbbb
cccccccc eeeeeeeeee XXX aaa bbbb cccc ddd
aaaa bbbb ddd ee XXX aaa bbb dddd eee

Output sample data:
aaaaaa bbbbbb cccc XXX_aaaaa_bbbbbbb
cccccccc eeeeeeeeee XXX_aaa_bbbb_cccc_ddd
aaaa bbbb ddd ee XXX aaa_bbb_dddd_eee

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