We prove that there is a Hermitian self-orthogonal$k$-dimensional truncated generalised Reed-Solomon code of length$n \leqslant q^{2}$over${\mathbb F}_{q^{2}}$if and only if there is a polynomial$g \in {\mathbb F}_{q^{2}}$of degree at most$(q-k)q-1$such that$g+g^{q}$has$q^{2}-n$distinct zeros. This allows us to determine the smallest$n$for which there is a Hermitian self-orthogonal$k$-dimensional truncated generalised Reed-Solomon code of length$n$over${\mathbb F}_{q^{2}}$, verifying a conjecture of Grassl and Rötteler. We also provide examples of Hermitian self-orthogonal$k$-dimensional generalised Reed-Solomon codes of length$q^{2}+1$over${\mathbb F}_{q^{2}}$, for$k=q-1$and$q$an odd power of two.