Polynomial solution conjecture for cubic sums and fourth or sixth powers

Let $k\in\{4,6\}$, and let $x,y,a,b$ be polynomials in one variable with integer coefficients and positive leading terms. The polynomials are coprime when they satisfy the coprimality condition intended in the source. A partial sum means one of the sums obtained from the expression below by successively combining its displayed terms.

Polynomial solution conjecture. For each choice of the sign, there is a solution of

$$x^3\pm y^3=a^k+b^k$$

in coprime polynomials $(x,y,a,b)$ such that no partial sum in

$$x^3\pm y^3-(a^k+b^k)$$

is a zero polynomial.

The conjecture is motivated by computational searches for coprime positive integer solutions, especially for $k=4$ and, less conclusively, for $k=6$. The supplied context says that the authors were unable to find a coprime polynomial solution, so the assertion remains open.