Singular simplicial sets that are (nerves of) categories

The simplicial sets that are both Kan complexes and nerves of a category are exactly the groupoids.

Simplicial sets of the form $\mathrm{Sing}_\bullet(X)$ are a proper subclass of Kan complexes.

Question: Can one characterize which simplicial sets of the form $\mathrm{Sing}_\bullet(X)$ are nerves of categories? (Either in terms of a condition on $X$ or a condition on the category $\mathcal C$, in the sense of "a simplicial set of the form $\mathrm{Sing}_\bullet(X)$ is the nerve of a category if and only if it is the nerve of a category $\mathcal C$ that satisfies $\mathrm{condition}(\mathcal C)$".)

Nerves of categories are 2-coskeletal, i.e. right orthogonal to the boundary inclusion $\partial \Delta^n \hookrightarrow \Delta^n$ for all $n > 2$. On the other hand, the singular set of a topological set $X$ is right orthogonal to $\partial \Delta^n \hookrightarrow \Delta^n$ if and only if $X$ itself is right orthogonal to the geometric realisation of $\partial \Delta^n \hookrightarrow \Delta^n$. This almost never happens. The existence of many continuous endomaps of $\left| \Delta^n \right|$ that restrict to the identity on $\left| \partial \Delta^n \right|$ means you can easily manufacture more continuous maps $\left| \Delta^n \right| \to X$ with the same restriction to $\left| \partial \Delta^n \right|$ – unless the map is constant on $\left| \Delta^n \right| \setminus \left| \partial \Delta^n \right|$.

Similarly, the existence of many continuous endomaps of $\left| \Delta^2 \right|$ that restrict to the identity on $\left| \Lambda^2_1 \right|$ makes it difficult for $X$ to be right orthogonal to the geometric realisation of $\Lambda^2_1 \hookrightarrow \Delta^2$, unless every continuous map $\left| \Delta^1 \right| \to X$ is constant. This implies every continuous map $\left| \Delta^n \right| \to X$ is constant, for all $n \ge 0$. Therefore:

Proposition. The singular set of $X$ is (isomorphic to) the nerve of a category if and only if, for every $n \ge 0$, every continuous map $\left| \Delta^n \right| \to X$ is constant.

For example, this happens if $X$ is discrete, or more generally, totally disconnected.