  Given a type A in homotopy type theory (HoTT), we can define the free
infinity-group on A as the loop space of the suspension of A+1. Equivalently,
this free higher group can be defined as a higher inductive type F(A) with
constructors unit : F(A), cons : A -> F(A) -> F(A), and conditions saying that
every cons(a) is an auto-equivalence on F(A). Assuming that A is a set (i.e.
satisfies the principle of unique identity proofs), we are interested in the
question whether F(A) is a set as well, which is very much related to an open
problem in the HoTT book. We show an approximation to the question, namely that
the fundamental groups of F(A) are trivial, i.e. that the 1-truncation of F(A)
is a set.
