Unsigned Bruhat edge sums need not square to zero

Given: The Axiom of Choice, the A2 setting $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_3$, simple roots ${\alpha }_1,{\alpha }_2$, $W=S_3=\{e,s_1,s_2,s_1{s}_2,s_2{s}_1,w_0\}$, a dominant integral weight $\lambda \in {\mathrm{\Lambda }}^{+}$, and the unsigned edge sums $d_k^{\mathrm{u}\mathrm{n}\mathrm{s}}\text{\hspace{0.17em}⁣}:C_k(\lambda )\to C_{k-1}(\lambda )$ whose $(w,w^{\mathrm{\prime }})$-component is ${\iota }_{w\to w^{\mathrm{\prime }}}$ for every cover $w⊳w^{\mathrm{\prime }}$ and $0$ otherwise (every sign $+1$).

$C_0(\lambda )=M(\lambda )$, $C_1(\lambda )=M(s_1\circ \lambda )\oplus M(s_2\circ \lambda )$, $C_2(\lambda )=M(s_1{s}_2\circ \lambda )\oplus M(s_2{s}_1\circ \lambda )$, and the covers of the A2 Bruhat graph are $w_0⊳s_1{s}_2$, $w_0⊳s_2{s}_1$, $s_1{s}_2⊳s_1$, $s_1{s}_2⊳s_2$, $s_2{s}_1⊳s_2$, $s_2{s}_1⊳s_1$, $s_1⊳e$, $s_2⊳e$ (The Bruhat graph and the BGG Verma sum in degree k, Bruhat intervals of rank two are diamonds).

For a cover $x⊳y$ the canonical embedding ${\iota }_{x\to y}\text{\hspace{0.17em}⁣}:M(x\circ \lambda )\hookrightarrow M(y\circ \lambda )$ is nonzero and injective; for a saturated path $x⊳m⊳y$ the composite ${\iota }_{m\to y}\circ {\iota }_{x\to m}$ is the canonical inclusion of $M(x\circ \lambda )$ into $M(y\circ \lambda )$ and is independent of the middle element $m$ (Bruhat covers give canonical Verma embeddings, and composites are inclusions, Dominant integral dot translates embed canonically in the Verma module).

The four nonzero components of $d_2^{\mathrm{u}\mathrm{n}\mathrm{s}}\text{\hspace{0.17em}⁣}:C_2\to C_1$ are the cover embeddings $s_1{s}_2\to s_1$, $s_1{s}_2\to s_2$, $s_2{s}_1\to s_1$, $s_2{s}_1\to s_2$, all with coefficient $+1$. The two nonzero components of $d_1^{\mathrm{u}\mathrm{n}\mathrm{s}}\text{\hspace{0.17em}⁣}:C_1\to C_0$ are $s_1\to e$ and $s_2\to e$, again with coefficient $+1$. Composition sums the component composites over intermediate summands (The BGG differential from signed Verma maps).