The commutativity conjectures for strong Lie algebras

Let $n$ be a positive integer, let $L$ be an $n$-strong Lie algebra over a field with at least $n$ elements, and let $A(L)$ be its universal enveloping algebra. For a monomial $p(x_1,\ldots,x_s)$, its multiweight records the degree in each variable. Strong-Lie-algebra commutativity conjectures. The following assertions should hold: (I) for all Lie elements $a,b\in A(L)$,

$$a^{n-1}b^{n-1}=(-1)^{n-1}b^{n-1}a^{n-1};$$

(II) for every monomial $p(x,y)$ of multiweight $(n-1,n-1)$ and all Lie elements $a,b\in A(L)$,

$$p(a,b)=(-1)^{n-1}p(b,a);$$

(III) for every monomial $p(x_1,\ldots,x_s)$ of multiweight $(n-1,\ldots,n-1)$, every collection of Lie elements $a_i\in A(L)$, and every permutation $\sigma$, the elements $p(a_1,\ldots,a_s)$ and $p(a_{\sigma(1)},\ldots,a_{\sigma(s)})$ are $\mathbb{F}_p$-linearly dependent. These conjectures seek a commutativity phenomenon in the enveloping algebras of strong Lie algebras; the source reports verification of the first identity for small values but leaves the general assertions conjectural.