Weak Lefschetz conjecture for ideals generated by powers and squarefree monomials

Let $R$ be the polynomial ring in $r$ variables over a field of positive characteristic, and let $I_{r,k,d}$ be the ideal

$$I_{r,k,d}=(x_1^k,\dots,x_r^k)+(\text{all squarefree monomials of degree }d).$$

Consider the Artinian algebra $R/I_{r,k,d}$. Weak Lefschetz conjecture. (a) If $d=4$, then $R/I_{r,k,d}$ has the WLP if and only if $k\bmod 4$ is $2$ or $3$. (b) If $d=5$, then the WLP fails. (c) If $d=6$, then the WLP fails. Here the WLP is the weak Lefschetz property for the algebra.