The conjectural Joyce structure from isomonodromic connections

Let $M$ be the moduli space of pairs $(C,Q_0)$, let $\mathcal O=\mathcal O(g,m)$ be the space of data $(C,Q_0,Q_1,Q_2)$ described above, and let $X^\#$ be the space equipped with the map $\alpha\colon\mathcal O\to X^\#$ and projection $\pi\colon X^\#\to M$. For each $\epsilon\in\mathbb C^*$, consider the projective-structure equation with parameter $\epsilon$ and its generalised monodromy. Joyce-structure conjecture. (i) For each $\epsilon\in\mathbb C^*$, there is a flat, meromorphic connection $g_\epsilon$ on $\pi\colon\mathcal O\to M$ whose leaves define deformations of the projective structure with constant generalised monodromy. (ii) There exist meromorphic connections $h_\epsilon$ on $\pi\colon X^\#\to M$ whose pullbacks via $\alpha\colon\mathcal O\to X^\#$ are the connections $g_\epsilon$ from (i). (iii) There is a meromorphic Joyce structure on $M$ obtained by combining the period structure and symplectic form on $M$ with the connections $h_\epsilon$ from (ii). The existence of the isomonodromy connections $g_\epsilon$ had not yet been established in the literature, so these assertions formulate the conjectural construction of the Joyce structure.