A Solution to Iima--Yoshino Problem 2.3

Iima and Yoshino asked for an ideal $I$ in $S=k[x_1,x_2,\ldots]$, with $\operatorname{deg} x_i=i$, and a monomial order such that $S/I\cong k[x_i:i\equiv\pm1\pmod5], \operatorname{in}(I)=(x_i^2,x_ix_{i+1}:i\geq1).$ We construct such an ideal and monomial order over every field $k$ of characteristic