In my previous question, I asked about nontrivial sums of four powers $a^m+b^n+c^k=d^l$, and whether the nature of the solutions depend on whether $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}+\frac{1}{l}$ is bigger, smaller, or equal to 1.

This heuristic (based on the Fermat Catalan conjecture, an analogous conjecture on three powers), was shown to have some rather glaring holes, with values of the reciprocal sum being arbitrarily close to 1/2 while still containing solutions. Therefore I propose a new question to repair the hole.

My new question is: Is it hard to find nontrivial integer coprime solutions to $\pm a^m\pm b^n\pm c^k=d^{lcm(m,l,k)}$ when $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}+\frac{1}{lcm(m,l,k)}\le1$? Are there $m,n,k$ for which it is known there are no solutions?

In my previous question, I asked about nontrivial sums of four powers $a^m+b^n+c^k=d^l$, and whether the nature of the solutions depend on whether $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}+\frac{1}{l}$ is bigger, smaller, or equal to 1.

This heuristic (based on the Fermat Catalan conjecture, an analogous conjecture on three powers), was shown to have some rather glaring holes, with values of the reciprocal sum being arbitrarily close to 1/2 while still containing solutions. Therefore I propose a new question to repair the hole.

My new question is: Is it hard to find nontrivial integer coprime solutions to $\pm a^m\pm b^n\pm c^k=d^{lcm(m,l,k)}$ when $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}+\frac{1}{lcm(m,l,k)}\le1$? Are there $m,n,k$ for which it is known there are no solutions?

Also, I would like to see parametric solutions for the cases where we do know solutions.