Boundary-zero conjecture for products of Eisenstein series

Let $E_k$ and $E_\ell$ be Eisenstein series, and define the weight-$k+\ell$ cusp form

$$\Delta_{k,\ell}=E_kE_\ell-E_{k+\ell},$$

with $k+\ell\geq16$ and $k\geq\ell$. The standard fundamental domain is bounded by the unit circle $|z|=1$ and the vertical lines $x=\pm1/2$.

Boundary-zero conjecture. All the zeros of $\Delta_{k,\ell}$ lying in the standard fundamental domain are on the boundary, $|z|=1$ or $x=\pm1/2$.

This conjecture contrasts with the expected equidistribution of zeros for random linear combinations of Hecke cusp forms. The supplied context does not state whether the conjecture has been proved or disproved.