The small-degree characteristic conjecture for the strong Lefschetz property

Let $R=K[x_0,\ldots,x_n]$ be a polynomial ring over a field $K$, let $I_{\underline{d}}=(x_0^{d_0},\ldots,x_n^{d_n})$, and let $t$ be the socle degree of $R/I_{\underline{d}}$. Assume that $d_0\leq\left\lceil\frac{t}{2}\right\rceil$. Small-degree characteristic conjecture. Then $R/I_{\underline{d}}$ has the strong Lefschetz property if and only if the characteristic of $K$ is either $0$ or greater than $t$. This is supported by the stated classification results, but the general assertion is presented as a conjecture and remains open.