The field-of-definition conjecture for differentially extremal forms

Let $f$ be a differentially extremal element of the algebra $\widetilde{M}$ of Drinfeld quasi-modular forms. A form is defined over a field $L\subset C$ if $f\in L[E,g,h]$, where $E$, $g$, and $h$ are the standard generators of $\widetilde{M}=C[E,g,h]$. Field-of-definition conjecture. If $f$ is differentially extremal, then $f$ is defined over $K$. The preceding lemma establishes this conclusion over the inseparable closure of $K$ in $C$ for normalised differentially extremal forms in $\widetilde{M}^{\leq 1}_{q^k+1,1}$; the conjecture asks for the stronger field of definition $K$ in general and is described as difficult with the available methods.