Initial minors conjecture for elliptic curve discrete logarithms

Fix a positive integer n^ and let k=3n^. Let $\mathcal{E}$ be an elliptic curve and let $P_1,\ldots,P_k$ be points on it. The Las Vegas reduction of the elliptic curve discrete logarithm problem produces a nonsingular matrix $\mathcal{A}$, and a set of initial minors is a set of minors such that every minor in the set is nonzero if and only if all minors of $\mathcal{A}$ are nonzero.

Initial minors conjecture. For a matrix $\mathcal{A}$ coming from the above approach to solve ECDLP, there exists a set of initial minors whose cardinality is bounded by a sub-exponential function of the size of the matrix.

If true, this would make the minor-testing step in the proposed Las Vegas algorithm substantially smaller than testing all minors, whose number is exponential in the matrix size. The supplied text does not state whether the conjecture has been proved or refuted.