The paucity problem for certain symmetric Diophantine equations

Let ${\phi }_1,\dots ,{\phi }_r\in \mathbb{Z}[z_1,\dots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text{deg}({\phi }_j)=k_j$ $(1\le j\le r)$, where $1\le k_1<k_2<\dots <k_r=k$. Subject to the condition $k_1+\dots +k_r\ge \frac{1}{2}k(k-1)+2$, we show that there is a paucity of non-diagonal solutions to the Diophantine system ${\phi }_j(\mathbf{x})={\phi }_j(\mathbf{y})$ $(1\le j\le r)$.