Conca–Krattenthaler–Watanabe power-sum regular-sequence conjecture

Let $S=\mathbb{C}[x_1,x_2,x_3]$. For positive integers $a,b,c$ with $a<b<c$, $\operatorname{gcd}(a,b,c)=1$, and $6\mid abc$, define

$$p_a(3)=x_1^a+x_2^a+x_3^a,\quad p_b(3)=x_1^b+x_2^b+x_3^b,\quad p_c(3)=x_1^c+x_2^c+x_3^c.$$

Conca–Krattenthaler–Watanabe's conjecture. The polynomials $p_a(3),p_b(3),p_c(3)$ form a regular sequence in $S$. Conca et al. verified some special cases, but the conjecture remains mysterious.