The four-variable power-sum regular-sequence conjecture from computer calculations

Let $p_i$ be the power sum symmetric polynomial of degree $i$ in $S=K[x_1,x_2,x_3,x_4]$. Let $A=\{a,b,n\}$ with $a<b<n$. The four-variable power-sum conjecture. The sequence $p_A(4)$ is regular if and only if the following conditions hold: (1) if $a$ is odd and $b$ is even, then this holds for every $n$; (2) if $a$ and $b$ are odd, then $n$ is even; (3) if $a=2m$ with $m$ odd, then it holds for every $n$ when $\lambda=b-a\ne4k$, while when $\lambda=4k$ it holds for every $n$ of the form $4l+2$; (4) if $a=2m$ with $m$ even, then $b\ne3a$ and $n\ne(2k+1)a$; and (5) $(a,b,n)$ is not of the form $(a,2a,5a)$. Here the parameters $k,l$ range over $\mathbb{N}$ as in the source. Computer calculations using CoCoA motivate this proposed classification, and the supplied text gives no proof or resolution.