Odd-variable equal-power weak Lefschetz conjecture

Let $R=k[x_1,\dots,x_{2n+1}]$ with $n\geq4$, let $L\in R$ be a general linear form, and set

$$I=\langle x_1^d,\dots,x_{2n+1}^d,L^d\rangle.$$

Odd-variable equal-power WLP conjecture. The ring $R/I$ fails the weak Lefschetz property if and only if $d>1$. Furthermore, when $n=3$, so that $R$ has seven variables, $R/I$ fails the WLP when $d=3$. This conjecture extends the proven seven-variable result for $d\geq4$ and records the unresolved general odd-variable case, with the degree-two seven-variable case known to have the WLP.