= Solution
The first derivative of a scalar is a covector, so
$$
\varphi_{;lm}=\partial_m\partial_l\varphi-\Gamma^p{}_{lm}\partial_p\varphi.
$$
Commuting partial derivatives and the <torsion-free connection> give \b[$\varphi_{;lm}=\varphi_{;ml}$], the <symmetric Hessian of a scalar field>. Its next <covariant derivative> is
$$
\varphi_{;lmn}=\partial_n(\varphi_{;lm})-\Gamma^p{}_{ln}\varphi_{;pm}-\Gamma^p{}_{mn}\varphi_{;lp}.
$$
In particular, <tensor index symmetrization> over $l,m$ adds nothing: $\varphi_{;(lm)n}=\varphi_{;lmn}$. This does not make the third index interchangeable with the first two. Applying the lower-index <Ricci identity> to the covector $\varphi_{;l}$ yields the requested <tensor index antisymmetrization>:
$$
\boxed{\varphi_{;l[mn]}=-\frac12R^p{}_{lmn}\varphi_{;p}.}
$$
Parentheses and brackets include the usual averaging factors. To make the remaining symmetry explicit, let $S_{lmn}=\varphi_{;(lmn)}$. Since the first two derivative indices are already symmetric,
$$
S_{lmn}=\frac13(\varphi_{;lmn}+\varphi_{;lnm}+\varphi_{;mnl}).
$$
The two commutator identities give $\varphi_{;lnm}=\varphi_{;lmn}+R^p{}_{lmn}\varphi_{;p}$ and $\varphi_{;mnl}=\varphi_{;lmn}+R^p{}_{mln}\varphi_{;p}$. Thus the <third covariant derivatives of a scalar> can also be written
$$
\boxed{\varphi_{;(lm)n}=S_{lmn}-\frac13\left(R^p{}_{lmn}+R^p{}_{mln}\right)\varphi_{;p}.}
$$
The right-hand side displays exactly the obstruction to total symmetry. At a point where the scalar gradient vanishes, or where these curvature contractions vanish, the third derivative is totally symmetric.
For the final contraction use $R_{ab}=R^c{}_{acb}$. The arbitrary-valence formula gives
$$
T^{ik}{}_{;ik}-T^{ik}{}_{;ki}
=R^i{}_{pik}T^{pk}+R^k{}_{pik}T^{ip}
=R_{pk}T^{pk}-R_{pi}T^{ip}
=R_{pk}(T^{pk}-T^{kp}).
$$
The <Ricci tensor> of the <Levi-Civita connection> is symmetric, whereas the last factor is antisymmetric. Their contraction is zero, proving
$$
\boxed{T^{ik}{}_{;ik}=T^{ik}{}_{;ki}.}
$$
No symmetry of $T^{ik}$ was assumed. This <double divergence of a contravariant tensor> identity uses the metric-compatible connection of <general relativity>: zero torsion alone for an arbitrary affine connection does not guarantee a symmetric <Ricci tensor> and would not suffice for this final step.
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