Pardue's Conjecture E on generic initial ideals

Let $K$ be a field, let $R=K[x_1,\ldots,x_n]$, and let $I=(f_1,\ldots,f_n)$ be a generic homogeneous ideal of type $(n;d_1,\ldots,d_n)$. Let $\operatorname{in}(I)$ be the initial ideal in the degree reverse lexicographic order. For a monomial $x^\alpha$, define $\max(x^\alpha)$ to be the largest index $i$ such that $x_i$ divides $x^\alpha$. Pardue's conjecture. If $x^\mu$ is a minimal generator of $\operatorname{in}(I)$ and $\max(x^\mu)=m$, then every monomial of the same degree in the variables $x_1,\ldots,x_{m-1}$ belongs to $\operatorname{in}(I)$. This conjecture is presented as a consequence of Moreno-Socías's conjecture and as equivalent, after Pardue's modification, to Fröberg's conjecture; its general resolution is not supplied here.