You tell me you've got everything you want, and Marianne Faithful can sing?

According to a TV show I’ve just been watching, the Beatles’ song “And Your Bird Can Sing” was a dig at Mick Jagger’s relationship with Marianne Faithful. If it’s true, it’s too wonderful for words – and has completely reinvigorated that song for me. But although it’s extremely entertaining, I can find no evidence for it on the internet. Please let it be true… Please!

Tell me that you’ve got everything you want
And your bird can sing
But you don’t get me
You don’t get me

You say you’ve seen seven wonders
And your bird is green
But you can’t see me
You can’t see me

When your prized possessions
Start to wear you down
Look in my direction
I’ll be round, I’ll be round

When your bird is broken
Will it bring you down?
You may be awoken
I’ll be round, I’ll be round

You tell me that you’ve heard every sound there is
And your bird can swing
But you can’t hear me
You can’t hear me