  We complete the program, initiated in [6], to compare the many different
possible definitions of the underlying homotopy type of a log scheme. We show
that, up to profinite completion, they all yield the same result, and thus
arrive at an unambiguous definition of the profinite homotopy type of a log
scheme. Specifically, in [6], we define this to be the profinite \'etale
homotopy type of the infinite root stack, and show that, over $\mathbb{C},$
this agrees up to profinite completion with the Kato-Nakayama space. Other
possible candidates are the profinite shape of the Kummer \'etale site
$X_{\mbox{k\'et}},$ or of the representable \'etale site of $\sqrt[\infty]{X}.$
Our main result is that all of these notions agree, and moreover the profinite
\'etale homotopy type of the infinite root stack is not sensitive to whether or
not it is viewed as a pro-system in stacks, or as an actual stack (by taking
the limit of the pro-system). We furthermore show that in the log regular
setting, all these notions also agree with the \'etale homotopy type of the
classical locus $X^{\mbox{triv}}$ (up to an appropriate completion). We deduce
that, over an arbitrary locally Noetherian base, the \'etale homotopy type of
$\mathbb{G}_m^N$ agrees with that of $B\mu_\infty^N$ up to completion.
