The symmetry-principle link-existence conjecture for height four complete intersections

Fix an integer $d$ and consider the sequence of residual schemes $Y_{i-1}$ arising in the paper's linkage strategy for a height four equigenerated complete intersection. For $3\leq i\leq d-2$, the $i$-th link is said to exist when the required forms can be chosen without a common factor in the initial degree of the ideal of $Y_{i-1}$. The Symmetry Principle is the geometric obstruction used in the paper: an excess concentration of points on a curve contradicts the required symmetric behavior of the parallel residual schemes.

Link-existence conjecture. For any $i$, $3\leq i\leq d-2$, if the $i$-th link does not exist, so that there is a common factor in the initial degree of the ideal of $Y_{i-1}$, then this common factor defines a curve containing enough points of $Y_{i-1}$ to yield a contradiction via the Symmetry Principle.

If true, this would establish the existence of all remaining links in the proposed proof strategy and, together with the exact-one failure conjecture, would imply the WLP in the relevant general setting. The paper leaves this technical point unresolved.