Positive-characteristic weak Lefschetz conjecture for Artinian monomial algebras of type two

Let $K$ be a field and let $R=K[x,y,z]$. An Artinian monomial ideal $I\subset R$ gives an Artinian monomial algebra $R/I$, whose type is the dimension of its socle. The algebra $R/I$ has the weak Lefschetz property if some linear form induces multiplication maps of maximal rank between consecutive graded components. Type-two positive-characteristic conjecture. If $R/I$ is of type two and has the weak Lefschetz property in characteristic zero, then it also has the weak Lefschetz property in characteristics

$$p>\frac{1}{2}(a+b+c).$$

Here $a,b,c$ are the exponents appearing in the preceding characteristic bound, where the source's supplied statement does not reintroduce their definition. The conjecture is motivated by computer experimentation and asserts that the preceding positive-characteristic bound is far from optimal; its resolution is not given in the source.