The 6abc regular-sequence conjecture for three power sums

Let $K$ be a field of characteristic zero, let $p_m=x_1^m+x_2^m+x_3^m$, and let $a,b,c$ be integers satisfying $0<a<b<c$ and $\operatorname{gcd}(a,b,c)=1$. The 6abc conjecture. The polynomials $p_a,p_b,p_c$ form a regular sequence if and only if $6$ divides $abc$. This conjecture is unresolved in general; one direction holds more generally, and the claim is known for certain special values, including $a=1$.