Generalization of a result of Hickel, Ito and Izumi about a Diophantine inequality

An important result of Ito and Izumi about a Diophantine inequality and the linearity of the Artin approximation function has been generalized recently by Hickel, Ito and Izumi for any excellent Henselian local domain. Our objective is to show that their results are also true in the case when R is an analytically irreducible Henselian local domain (without the excellence hypothesis).