Are functions under which preimage of recursive sets are recursive sets always recursive?

Question

It is well known that preimage of a recursive set under a recursive function is a recursive set (see for example Image and preimage of a recursive function on a recursive set).

I wonder whether some converse is true:

Assume the preimage of every recursive set under a given function from $\Bbb N$ to $\Bbb N^k$ is a recursive set.

Is this function necessarily recursive ?