Infinitude conjecture for cubic sums and fifth powers

Let $a,b,x,y$ be positive integers, and let each occurrence of the sign $7$ be chosen consistently. A solution is nontrivial in the sense intended by the Diophantine equation under consideration.

Infinitude conjecture. For each choice of the signs, the equation

$$x^3\pm y^3=a^5+b^5$$

has infinitely many nontrivial solutions in positive integers.

This conjecture is motivated by the authors' numerical search; the surrounding results establish analogous infinitude statements for some other exponents, but do not resolve the exponent $5$ case.