Do the squares contain arbitrarily long quasi-progressions and arbitrarily large cubes? (Erdős #782)

An increasing sequence $x_1<\cdots<x_k$ is a quasi-progression with bound $C$ if there is a common difference $d$ with $x_i+d\le x_{i+1}\le x_i+d+C$ for all $1\le i<k$. An (affine/Hilbert) cube of dimension $r$ is a set $a+\big\{\sum_{i=1}^r\epsilon_i b_i:\epsilon_i\in\{0,1\}\big\}$ with $b_i>0$. Brown, Erdős and Freedman asked: (1) is there a constant $C>0$ such that for every $k$ the set of perfect squares contains a $k$-term quasi-progression with bound $C$; and (2) do the squares contain arbitrarily large (high-dimensional) cubes? An affirmative answer to (1) implies an affirmative answer to (2).

Acceptance. FULLY RESOLVES: a complete proof (Lean/Coq preferred, else full written). AFFIRMATIVE: a construction/argument producing, for a fixed $C$, arbitrarily long quasi-progressions of squares (resp. arbitrarily high-dimensional cubes of squares), with proof. NEGATIVE: a proof that for every $C$ the length of quasi-progressions of squares is bounded (resp. the dimension of cubes of squares is bounded) — an unconditional version of the Cilleruelo–Granville result qualifies. ADVANCES: a proven unconditional upper bound on the maximal dimension $r$ of a cube contained in the squares improving on the unconditional $r\le 7\log\log N$ of Dietmann–Elsholtz (stated in the background), or an unconditional absolute (finite) bound improving on the Bombieri–Lang-conditional finiteness; a proven bound on quasi-progression length as a function of $C$; or a reproducible exhaustive search exhibiting a record-length quasi-progression of squares at a stated $C$, or a record-dimension cube of squares, with the search code and a certificate of the squares involved. Deliver the proof, the improved bound with proof, or the search code plus the record configuration and its verification.