What are the values of $w$ such that the corresponding $\text{Keccak-}f[25w]$ function is invertible (bijective)? From this answer, I see that from a mathematical point of view, one could define the $\text{Keccak-}f[25w]$ permutation for an arbitrary value of $w$.  

I am interested in the following question: disregarding all properties of the resulting function except the invertibility (because $\text{Keccak-}f[25w]$ is required to be a permutation), is it true that if the value of $w$ is a multiple of $8$ and not a multiple of $5$, the corresponding $\text{Keccak-}f[25w]$ function is invertible (and bijective)? If the answer is “No”, then how to determine whether a particular $w$ is suitable (assuming that $w$ is required to be a multiple of $8$)?
$\operatorname{Keccak-}\!f[25w]$ is defined, by the Keccak specification, only when $w = 2^\ell$ for some $\ell$, and in all such cases it is a permutation.  The choices 1, 2, and 4 for $w$ are not multiples of 8.  No multiple of 5 is a power of 2 and consequently $w$ is never a multiple of 5.