= Solution
Use the connection convention $\nabla_\alpha Y^\rho=\partial_\alpha Y^\rho+\Gamma^\rho{}_{\mu\alpha}Y^\mu$, with the derivative index last. This is consistent with the printed curvature formula and the final formula in part (iv). The paper's $T_{\mu\nu}{}^\rho=\Gamma^\rho{}_{\mu\nu}-\Gamma^\rho{}_{\nu\mu}$ is then the negative of the geometric <torsion tensor> defined by $\mathcal T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]$. Keep the paper's component convention throughout this question.
Set $A_{\mu\nu\rho}=g_{\rho\lambda}(\Gamma^\lambda{}_{\mu\nu}-S^\lambda{}_{\mu\nu})$. The <difference of affine connections is a tensor>. Since the <Levi-Civita connection> $S$ is symmetric and metric compatible, expanding the <nonmetricity tensor> gives
$$
T_{\mu\nu\rho}=A_{\mu\nu\rho}-A_{\nu\mu\rho},\qquad
N_{\mu\nu\rho}=-A_{\mu\rho\nu}-A_{\nu\rho\mu}.
$$
Here $T_{\mu\nu\rho}=g_{\rho\lambda}T_{\mu\nu}{}^\lambda$ is antisymmetric in its first two slots, while $N_{\mu\nu\rho}$ is symmetric in its first two slots. Cyclically permuting these identities and eliminating the other components gives
$$
A_{\mu\nu\rho}=\frac12\left(T_{\mu\nu\rho}+T_{\nu\rho\mu}-T_{\rho\mu\nu}
+N_{\mu\nu\rho}-N_{\nu\rho\mu}-N_{\rho\mu\nu}\right).
$$
Raise the last slot to obtain \b[the connection decomposition]
$$
\boxed{\Gamma^\rho{}_{\mu\nu}=S^\rho{}_{\mu\nu}-K_{\mu\nu}{}^\rho+W_{\mu\nu}{}^\rho,}
$$
where
$$
K_{\mu\nu}{}^\rho=-\frac12\left(T_{\mu\nu}{}^\rho+T_\nu{}^\rho{}_\mu-T^\rho{}_{\mu\nu}\right),\qquad
W_{\mu\nu}{}^\rho=\frac12\left(N_{\mu\nu}{}^\rho-N_\nu{}^\rho{}_\mu-N^\rho{}_{\mu\nu}\right).
$$
\b[The PDF is missing the factor $1/2$ in its definition of $K$.] With the printed $K$, the decomposition would instead contain $-K/2$. In these formulas $T^\rho{}_{\mu\nu}$ raises the first slot of the fully lowered <tensor>; it must not be confused with $T_{\mu\nu}{}^\rho$, which raises the last slot.
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