The characteristic-two conjecture for the weak Lefschetz property of monomial complete intersections

Let $K$ be a field of characteristic $2$, let $d$ be a positive integer, and set

$$A=K[x,y,z]/(x^d,y^d,z^d).$$

The characteristic-two conjecture. The algebra $A$ has the weak Lefschetz property if and only if

$$d=\left\lfloor \frac{2^n+1}{3}\right\rfloor$$

for some positive integer $n$.

The conjecture gives an explicit characterization of the equal-exponent monomial complete intersections having the weak Lefschetz property in characteristic $2$; in particular, since the listed values of $d$ are odd, it predicts that $A$ has the weak Lefschetz property for every even $d$.