Let $q$ be a power of an odd prime $p$. We prove the MDS conjecture over $\mathbb F_q$ in dimension $k$ whenever \[ 2\leqslant k\leqslant B(p,q) \quad\text{or}\quad q+2-B(p,q)\leqslant k\leqslant q, \] where $B(p,q)=\left\lfloor\frac{(p-2)q+6p-10}{2p-3}\right\rfloor$. For fixed $p$, this gives a range of dimensions linear in $q$, in contrast with the square-root scale of the previously known general unconditional ranges over proper extension fields, and proves the conjecture in an asymptotic proportion $1-1/(2p-3)$ of all dimensions. The main structural ingredient is a full-support obstruction for determinant relations on arcs over arbitrary fields of odd characteristic. After projection from $p-1$ points, local matching constraints followed by multilinear descent force the resulting homogeneous system to have zero kernel. For a hypothetical $q+2$-point arc over $\mathbb F_q$, the Ball--Lavrauw construction produces precisely the forbidden relations. The same framework also proves Chowdhury's conjecture that a family of inclusion matrices has full row rank.

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