Period-preserving involution conjecture for elliptic surfaces with $p_g=q=1$

Let $F_{1,1}$ denote the moduli space of elliptic surfaces with $p_g=q=1$. For a surface $S$ with base curve $C$ and canonical fiber $F$, use primes to denote the corresponding objects for a second surface $S'$.

Period-preserving involution conjecture. The space $F_{1,1}$ admits a period-preserving birational involution $S\leftrightarrow S'$ such that

$$j(C)=j(F')\quad\text{and}\quad j(F)=j(C').$$

Moreover, $S$ and $S'$ are moduli spaces of stable vector bundles on each other of rank $2$, determinant $\mathcal O(s)$, and second Chern class $c_2=\operatorname{pt}$; a Fourier–Mukai transform induces an isomorphism of their integral Hodge structures.

The conjecture is motivated by two degeneration constructions whose period images dominate the same boundary divisor. It proposes a strong symmetry exchanging the base and canonical-fiber data, together with a moduli-theoretic and Hodge-theoretic realization; no resolution is given here.