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(71,382) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{4}$;

(211,382) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{4}$;

(71,67.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{4}$;

(192,67.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{3}$;

(381,452.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{5}$;

(561,452.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{5}$;

(381,17.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{5}$;

(542,17.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $x_{4}$;

Induced morphisms for paths in $\Gamma_{1234321}$ and $\Gamma_{123454321}$.

There are also elements of symmetric groups which can be used to produce counterexamples, and whose conflated expression graphs are not linear, i.e., elements whose conflated expression graph is different from Figure <ref>. The reader can verify that one such element is $12134325$ in $S_6$.

Note that all elements in our family of counterexamples are paths with $p_a\notin \{\mathbf{s}, \mathbf{t}\}$ and $p_z\notin \{\mathbf{s}, \mathbf{t}\}$. This could make us think that the behavior for complete paths with $p_a\in \{\mathbf{s}, \mathbf{t}\}$ or $p_z \in \{\mathbf{s}, \mathbf{t}\}$ is different. That is not the case! Consider the same element $\sigma=12321$, and $\Gamma_{\sigma}$, the paths $[\mathbf{s},c,\mathbf{t},c,\mathbf{s},c]$ and $[\mathbf{s},c,\mathbf{t},c]$ give a counterexample.
It is enough to evaluate both path morphisms in the element
$$1 \otimes_{s_1} x_2 \otimes_{s_2} 1 \otimes_{s_3} 1 \otimes_{s_2} 1 \otimes_{s_1} 1.$$

These counterexamples (and some others that we do not show here) have in common particular choices of elements and paths, however, verification for other families of elements show that there is a phenomenon hidden underneath. To be precise, we have observed that for the longest element in $S_n$, any path $p$ with $p_a=\mathbf{s}$ and $p_z=\mathbf{t}$ will be equivalent to an oriented path from $\mathbf{s}$ to $\mathbf{t}$ (we proved this for $S_4$ in Section <ref>). In other words, we do not need to follow the Manin-Schechtman orientation as long as we start and end in the right vertices. The same when we start in $\mathbf{t}$ and end in $\mathbf{s}$. We propose the following strengthening of the FPC for $w_0$.

Let $w_{0,n} \in S_n$ and $\mathbf{s}$, $\mathbf{t}$ be the source and sink of the Manin-Schechtman orientation. Let $p, q$ be two paths in $\Gamma_{w_{0,n}}$, which pass through $\mathbf{s}$ and $\mathbf{t}$, satisfying $p_a=q_a$, and $p_z=q_z$. Then $f(p)=f(q)$.

We also conjecture the same for other choices of $\mathbf{s}$ and $\mathbf{t}$ obtained from other orientations different from the one given by Manin and Schechtman. Of course, this conjecture implies the FPC for $w_{0,n}$.

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