The four-power-sum regular-sequence conjecture

Let $K$ be a field of characteristic zero, let $S=K[x_1,x_2,x_3,x_4]$, and write $p_m=x_1^m+x_2^m+x_3^m+x_4^m$. Let $A=\{a_1,a_2,a_3,a_4\}$ be a set of positive integers with $\operatorname{gcd}(a_1,a_2,a_3,a_4)=1$, and let $\nu_2$ denote the $2$-adic valuation. The four-power-sum conjecture. The polynomials $p_{a_1},p_{a_2},p_{a_3},p_{a_4}$ form a regular sequence if and only if all of the following hold: $24$ divides $a_1a_2a_3a_4$; the set $\{\nu_2(a_i)\mid a_i\in A\}$ contains at least two distinct positive integers; and $A$ contains no subset of the form $\{d,2d,5d\}$ for any $d\in\mathbb{N}$. The conditions are stated to be necessary and independent, but the conjecture is not resolved in the source.