Higher Nash blowups of normal toric varieties in prime characteristic

We prove that the higher Nash blowup of a normal toric variety defined over a field of positive characteristic is an isomorphism if and only if it is non-singular. We also extend a result by R. Toh-Yama which shows that higher Nash blowups do not give a one-step resolution of the $A_3$-singularity. These results were previously known only in characteristic zero.