Before revision:  Recently (arXiv 2021) , Alman proved that for any field \mathbb{F, d\geq 2 and arbitrary }  matrices M_1, \ldots, M_n \in \mathbb{F^{d\times d}, one can get a d^n\times d^n matrix of rank } d^{(1-\gamma)n \dots\ge  ^{ ( 1+\varepsilon ) n } entries of the Kronecker product M = M_1\otimes M_2\otimes \ldots\otimes M_n\le  , where 1/\gamma is roughly 2^d/\varepsilon^2.
After revision:  Recently (arXiv 2021) , Alman proved that for any field \mathbb{F, d\geq 2 and arbitrary } matrices M_1, ^{d\times d}, one can get a d^n\times d^n matrix of rank } d^{(1-\gamma)n \dots\ge ^{ ( 1+\varepsilon ) n } entries of the Kronecker product M = M_1\otimes M_2\otimes \ldots\otimes M_n\le , where 1/\gamma is roughly 2^d/\varepsilon^2.