The vanishing WLP probability conjecture for random Artinian monomial algebras

Let $S$ be a polynomial ring in $n$ variables, let $I\sim\mathcal{I}(n,D,p)$, and consider the random Artinian algebra

$$A=S/\bigl(I+(x_1^D,\dots,x_n^D)\bigr),$$

where $n\geq 3$. Set $p=1/D^2$. The vanishing WLP probability conjecture. As $D$ grows, the probability that $A$ has the weak Lefschetz property tends to zero:

$$\lim_{D\to\infty}\mathbb{P}(A\text{ has the WLP})=0.$$

The conjecture is motivated by simulations showing that failure of the weak Lefschetz property is particularly common at $p=1/D^2$. The paper provides experimental evidence but no proof of the stated asymptotic claim.