Let $F$ be a free group of finite rank with a fixed basis and corresponding word metric. Let $Q = Q^0_h(F, \mathbb{R})$ be the space of real homogenous quasimorphisms that vanish on the basis of $F$. In Grigorchuk's "Some results on bounded cohomology", he gives a description of $Q$ as follows.
Let the length of a conjugacy class of $F$ be the word-length of a shortest representative. Then there exists a set $P$ of shortest representatives of conjugacy classes of simple elements (i.e., not proper powers) of length at least 2, such that $P = P^{-1}$ and every element of $P$ is a non-overlapping word: that is, a word $W$ that cannot be written as a reduced word $W = xyx$. Let $P^+$ be a subset containing only one of $W, W^{-1}$ for each $W \in P$. Let $e_W$ be the Brooks quasimorphism corresponding to $W$. Then any element of $Q$ can be written uniquely as: $$ \sum\limits_{W \in P^+} \alpha_W e_W.$$ (Here we are working with pointwise convergence).
Now, in the same article, the author ends by saying that it is a question of some interest to understand which sequences $(\alpha_W)_{W \in P^+}$ give rise to elements of $Q$. He proves that all $\ell^1$ sequences do, but gives an example showing that this is not a necessary condition.
My question is: has there been any further work on this? Is there a complete characterization of which sequences work, or just any further result in this direction?
