Let $M_i$ be fixed rectangular matrices with integer coefficients less than $n$. Consider the variety defined by the condition $$ \mathrm{rank}(\lambda_1M_1 + \lambda_2M_2 + ... + \lambda_kM_k) = 1. $$

Assume that there is an integral point $(\lambda_1, ..., \lambda_k)$ lying on this variety, can we then guarantee that there is an integral point with the height (maximal absolute value of the coordinates) bounded polynomially in $n$ (the exponent may depend on $k$ and the size of the matrices)?

P. S.

It might be related to the Birch-Davenport theorem, so with high probability this question is either known or out of reach for current methods (which would be ok to know), but I couldn't find a reference.