In this work, we consider the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. It has been previously conjectured that for any given $k times n$ binary matrix $M$, if $M$ does not have an all-zero $a times b$ sub-matrix such that $a+b>k$, then there exists a $k times n$ matrix $G$, with entries in the field $ mathbb{F}_q$ for any $q geq n+k−1$, such that $G$ generates an $[n,k]_q$ MDS code and $G$ has the support $M$. This conjecture is known as the emph{GM-MDS conjecture}. Despite all the attempts by the coding theory community, this conjecture remains still open in general. In this work, we propose a novel technique for proving a stronger algebraic-combinatorial conjecture (connecting the Wronskian-type determinant of a family of sets and their pattern of intersections), and demonstrate the strength of our technique by proving the GM-MDS conjecture for several non-trivial and near-extremal cases of $M$.