The optimal Sturm bound conjecture for quasimodular forms

Let $k\geq 1$ be an integer, let $\widetilde{M}_{2k}$ be the space of quasimodular forms of weight $2k$, and let $f\in\widetilde{M}_{2k}$. The Fourier coefficients of $f$ are the coefficients in its $q$-series expansion.

Optimal Sturm bound conjecture. The form $f$ is determined uniquely by its first $\dim \widetilde{M}_{2k}$ Fourier coefficients. Furthermore, for every integer $m\geq 2$, the reduction $f\pmod{m}$ is uniquely determined by its first $\dim \widetilde{M}_{2k}$ Fourier coefficients modulo $m$.

Computations for $2\leq k\leq 20$ support the conjecture, which would establish that the theoretically optimal Sturm bound holds for quasimodular forms. The claim remains open in the source.