Solymosi's affine-cube conjecture for squares

An affine cube of dimension $d$ in $\mathbb Z$ is a set

$$\left\{b_0+\sum_{i\in I}b_i:I\subset\{1,\ldots,d\}\right\},$$

where $b_0,\ldots,b_d$ are nonzero integers. Solymosi's conjecture. There exists an integer $d>0$ such that no affine cube of dimension $d$ consists of distinct squares. The source explains that this follows from the Bombieri--Lang conjecture, but is not known unconditionally.