The no-pinned-entry construction conjecture for even-order latin squares

Let $n\geqslant 28$ be an even integer. A latin square of order $n$ is an $n\times n$ array in which each symbol occurs exactly once in each row and column, and a transversal is a selection of $n$ entries containing one representative of every row, column, and symbol. An entry is pinned if it belongs to every transversal. No-pinned-entry construction conjecture. There is a latin square of order $n$ that has no two disjoint transversals, but also has no pinned entry. The claim gives even-order latin squares whose transversals pairwise intersect without all sharing one entry; the paper states that its constructions support the conjecture, but that a general argument guaranteeing the required transversals is not known.