Let $S$ be a proper subset of $\{1,\ldots,n\}$ and consider a function
$$f(x_1,\ldots,x_n) = \log \sum_{i=1}^n e^{x_i} - \log \sum_{i \in S} e^{x_i},$$ so that the first term is a log-sum-exp function of n variables, and the second term is a log-sum-exp function of a subset of this variables.
Is this function convex? It is the difference of two convex functions, and it seems that the first term should be ``more convex'' than the second one, but the Hessian gets very messy.
Consider $n=3$ and $S=\{1,2\}\subset\{1,2,3\}$. If the corresponding $f$ is convex, then
$$ t \mapsto f(t, -t, 0) = \log\left(1+\frac{1}{2\cosh t}\right) $$
should be also convex. However, this functions is not convex. (Note that this function achieves the maximum value at $t = 0$ and then decays to $0$ as $t \to \pm \infty$.)