= Solution
Choose a basis of $T_pM$ and use the <affine exponential map> to define <affine normal coordinates>. Its differential at zero is the identity, so the <inverse function theorem> gives a chart near $p$. By part (iii), radial <geodesics> have coordinates $x^\mu(s)=sX^\mu$. Their geodesic equations at $p$ imply $\Gamma^\lambda{}_{\mu\nu}(p)X^\mu X^\nu=0$ for every $X$. The <polarization identity> therefore gives $\Gamma^\lambda{}_{(\mu\nu)}(p)=0$.
The <disformation tensor> $W_{\mu\nu}{}^\lambda$ is symmetric in $\mu,\nu$, whereas $T_{\mu\nu}{}^\lambda$ is antisymmetric. Also $T_\nu{}^\lambda{}_\mu=-T^\lambda{}_{\nu\mu}$. Taking the symmetric part of the corrected <affine connection decomposition> consequently gives \b[the normal-coordinate identity]
$$
\boxed{S^\lambda{}_{\mu\nu}+W_{\mu\nu}{}^\lambda-\frac12T^\lambda{}_{\mu\nu}-\frac12T^\lambda{}_{\nu\mu}=0\quad\text{at }p.}
$$
Thus the final identity in the PDF is correct after the half-factor repair in part (i). In general $\Gamma^\lambda{}_{\mu\nu}(p)=\tfrac12T_{\mu\nu}{}^\lambda(p)$ in the paper's convention, rather than zero. Neither $S$ nor the first derivatives of the <metric tensor> need vanish in these <affine normal coordinates>. With the <Levi-Civita connection>, both torsion and nonmetricity vanish and the usual vanishing of the <Christoffel symbols> is recovered.
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