Lander–Parkin–Selfridge Sidon-set conjecture for sufficiently large powers

For a positive integer $k$, let $S_k=\{n^k:n\in\mathbb{Z}_{>0}\}$; a set is Sidon if there are no nontrivial equal sums of two of its elements. Lander–Parkin–Selfridge's special-case conjecture. There is an integer $k_0$ such that whenever $k\geq k_0$, there do not exist positive integers $a_1,a_2,b_1,b_2$ with

$$a_1^k+a_2^k=b_1^k+b_2^k$$

and $\{a_1,a_2\}\ne\{b_1,b_2\}$. Equivalently, for sufficiently large $k$, the set of $k$-th powers is Sidon.

This is explicitly identified as the special case of the preceding Lander–Parkin–Selfridge conjecture needed in the paper. It is used to obtain a stronger bound for Hilbert cubes in perfect powers.