The prime-ideal conjecture for pairs of power sums in four variables

Let $S=\mathbb{C}[x_1,x_2,x_3,x_4]$. The prime-ideal conjecture for pairs of power sums. The following assertions hold: (1) if $a$ is prime, $a\ge5$, and $b=a+m+6d$ with $m\in\{1,5\}$ and $d\in\mathbb{N}\cup\{0\}$, then $(p_a,p_b)$ is a prime ideal; consequently, $p_a,p_b,p_n$ is a regular sequence for every $p_n\notin(p_a,p_b)$; (2) if $m\in\mathbb{N}$ and $m\ne2+3k,2+4k$ for $k\in\mathbb{N}$, then $(p_2,p_m)$ is prime, with the analogous regular-sequence consequence; (3) if $m\in\mathbb{N}$ and $m\ne6+9\lambda$ for $\lambda\in\mathbb{N}\cup\{0\}$, then $(p_3,p_{2m})$ is prime, with the analogous consequence; and (4) if $m\in\mathbb{N}$ and $m\ne4+3k,4+8k$ for $k\in\mathbb{N}$, then $(p_4,p_m)$ is prime, with the analogous consequence. The authors state that these claims were suggested by computer experiments and that they could not prove them; their resolution is therefore open in the supplied text.