Question about A072087

[David Wilson]

Recently, I posted a proof that the iterates of a fully multiplicative 
function are fully multiplicative.  This is an almost trivial consequence of 
the fact that the fully multiplicative functions are closed under 
composition (an easy proof).  Duh.

Now I am faced with A072087, which is the least inverse of A072084.  I am 
tempted to think that the least inverse of any fully multiplicative function 
is fully multiplicative (where defined), but I'm a little to dizzy to come 
up with a proof.  Can anyone supply?