I'm not quite convinced. If you look at https://help.ubuntu.com/community/RemoveOldKernels you'll see that unattended-upgrades, which is installed by default, is going to purge old kernels for you, by default, but they must be marked as automatically installed.
The second step is to edit old configuration file to enable automatic removal, or preferably create an overriding new configuration file where you need to write only options that you change from the default or from the values defined in the shipped one
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