The complete-symmetric-polynomial regular-sequence conjecture for three variables

Let $h_k(3)$ be the complete symmetric polynomial of degree $k$ in three variables, and for $A=\{a,b,c\}$ with $a<b<c$ let $h_A(3)$ denote the set of polynomials $h_a(3),h_b(3),h_c(3)$. The complete-symmetric-polynomial regular-sequence conjecture. The set $h_A(3)$ is a regular sequence if and only if all of the following conditions hold:

1. $abc\equiv0\pmod{6}$.
2. $\operatorname{GCD}(a+1,b+1,c+1)=1$.
3. For every $t\in\mathbb{N}$ with $t>2$, there exists $d\in A$ such that $d+2\not\equiv0,1\pmod{t}$.

This conjecture proposes a complete arithmetic characterization of regular sequences formed by three complete symmetric polynomials in three variables; the source presents it without claiming a proof.