I originally posted this question over at Stackexchange, before realising it is much better suited for Overflow:

Let $\langle \; S \; | \; R \; \rangle$ be a presentation of a group $G$ with a set $R = R^{-1}$ of freely and cyclically reduced relators, and let $\Lambda$ be the girth of $\langle \; S \; | \; R \; \rangle$. Suppose that

1) the set $R$ of relators contains no proper power, and

2) for any triple $x,y,z \in F(S)$ of reduced words, such that $yx$ and $zx$ are two distinct (reduced) cyclic conjugates of elements of $R$, we have $$|x| \leq 1/6 \Lambda,$$ where $|x|$ denotes the length of $x$.

Then, the famous **Cancellation Theorem** says that the corresponding *presentation complex* $P(S,R)$ is aspherical.

Does anybody know if (and why) the constant $1/6$ in the statement is sharp ?

To be more precise, I am not looking for an aspherical presentation that fails to meet condition $2$ (such are easy to construct). I am looking for a group with a non-aspherical presentation (in the above sense) that satisfies both conditions, but with constant $c > 1/6$.

Ideally, I am looking for an answer to the following question: For every $\epsilon > 0$, does there exist a presentation $P_{\epsilon}$ of a group $G_{\epsilon}$, satisfying both conditions, but with constant $1/6 + \epsilon$, such that $P_{\epsilon}$ is not aspherical ?