Pollack's lifting conjecture for cusp-form relations

Let $k$ be an integer, and let $f$ be a cusp form of weight $k$ for $\operatorname{SL}_2(\mathbb{Z})$. Let $\mathrm{R}_{f,d}$ denote the relation associated with $f$ in the Lie algebra generated by the elements $\varepsilon_{2i}$. A bracket is an iterated Lie bracket of such elements.

Pollack's lifting conjecture. There exists a linear combination $T$ of brackets of $\varepsilon_{2i}$, containing at least three $\varepsilon_{2i}$ with $i>1$, such that

$$\mathrm{R}_{f,d}=T.$$

This conjecture asks when relations in the quotient by $\Theta^3\mathscr{E}$ lift to actual relations in $\mathscr{E}$. The source motivates it from examples associated with cusp forms, while noting that analogous relations associated with certain Eisenstein series do not lift in general.