Fintushel--Stern knot surgery symplecticity conjecture for torus Lefschetz fibrations

Suppose that $X^4$ is a closed $4$--manifold admitting a Lefschetz fibration whose regular fibers are tori. Let $T\subset X$ be a regular fiber of the fibration, and suppose that $[T]\ne0$ in $H_2(X;\mathbb R)$. Hence $X$ is symplectic. Let $X_K$ be a manifold obtained by Fintushel--Stern knot surgery on $T$ using a knot $K\subset S^3$. Fintushel--Stern knot surgery symplecticity conjecture. $X_K$ has a symplectic structure if and only if $K$ is a fibered knot. This conjecture concerns when knot surgery preserves symplecticity; the supplied text gives no resolution beyond the known construction of a symplectic structure when $K$ is fibered and the Seiberg--Witten obstruction for knots with non-monic Alexander polynomial.