The symplectic invariance and Taubes correspondence conjecture for surface counts

Let $(X,\omega)$ be a symplectic four-manifold and let $f$ be a Lefschetz pencil of sufficiently high degree on $X$. The invariant $\mathcal{I}_{(X,f)}$ counts sections of a compactified bundle of symmetric products and is defined on $H_2(X;\mathbb{Z})$, while $Gr_X$ denotes Taubes's Gromov invariant.

Symplectic invariance and Taubes correspondence conjecture. Once $f$ is of sufficiently high degree, $\mathcal{I}_{(X,f)}$ should be independent of $f$ and define a symplectic invariant of $(X,\omega)$. Moreover, there should be an equality

$$\mathcal{I}_{(X,f)} = Gr_X: H_2(X;\mathbb{Z}) \rightarrow \mathbb{Z}.$$

This would identify the surface-counting invariant obtained from Lefschetz pencils with Taubes's Gromov invariant and show that the former depends only on the symplectic structure, rather than on the chosen sufficiently high-degree pencil. The surrounding results establish related comparisons, including equality modulo two under a no-square-zero-symplectic-torus hypothesis, while the integral identification and full pencil independence remain conjectural here.