Newton polygons and the constant associated with the Prouhet–Tarry–Escott problem

In a 2017 article Filaseta and Markovich obtained new information on the lower bounds of $2$-adic valuation of certain constants $\overline {C_n}$ associated with the Prouhet–Tarry–Escott (PTE) problem for the cases $n=8$ and $n=9$ by using the classical theory of Newton polygons, and also pointed out that it would be of interest to obtain improved lower bounds in the cases when $10 \leq n \leq 12$. In the present article, we obtain new $2$-adic information on the lower bounds of $\overline {C_n}$ for the cases $n=10$ and $n=12$.