Chain-homotopy conjecture for the maps $g_1^j$ and $g_2^j$ on dihedral rack complexes

Let $R_q$ be the dihedral quandle of prime order $q>3$, and let $g_1^j,g_2^j:C_n^R(R_q)\to C_n^R(R_q)$ be the chain maps defined by

$$g_1^j(x)=\sum_{y\in R_q}(y,\ldots,y,y*x_j,x_{j+1},\ldots,x_n),$$ $$g_2^j(x)=\sum_{y\in R_q}(y,\ldots,y,x_j*y,x_{j+1},\ldots,x_n).$$

Chain-homotopy conjecture. The maps $g_1^j$ and $g_2^j$ are chain homotopic. The source notes that these maps are chain maps for the stated prime orders; proving their chain homotopy would complete the cancellation needed for the broader torsion argument.