Span conjecture for diagonal restrictions of Hilbert theta functions

Let $F$ be a totally real field of degree $d$, and let ${\mathcal{M}}_F^\theta$ be the subalgebra generated by theta functions associated with totally positive definite, $\mathbb{Z}$-unimodular ${\mathcal{O}}_F$-lattices of rank $8m$. Write $({\mathcal{M}}_F^\theta)^\Delta$ for its image under diagonal restriction, where $f^\Delta(\tau)=f(\tau,\ldots,\tau)$. The theorem established for $d=2,3$ gives

$$({\mathcal{M}}_F^\theta)^\Delta={\mathcal{M}}_\mathbb{Q}^{(4d/d_2)},\qquad d_2=\gcd(2,d).$$

Span conjecture. Equation $({\mathcal{M}}_F^\theta)^\Delta={\mathcal{M}}_\mathbb{Q}^{(4d/d_2)}$ holds for any totally real field $F$ of degree $d$. This predicts that diagonal restrictions of Hilbert theta functions generate the full algebra of elliptic modular forms of weight divisible by $4d/d_2$. The statement is proved in the paper for degrees $2$ and $3$; its validity for arbitrary totally real fields remains open.