Hadamard-product convexity conjecture for the class $\Omega$

Let $\Omega$ be the class of normalized analytic functions introduced in the paper, and let $*$ denote the Hadamard product. Let $\mathcal{C}$ denote the class of convex functions.

Hadamard-product convexity conjecture. For any $f_1,f_2\in\Omega$, one has

$$f_1*f_2\in\mathcal{C}.$$

This conjecture concerns the relationship between the class $\Omega$ and the convex class $\mathcal{C}$ under Hadamard products. The paper notes that $\Omega$ is closed under Hadamard product and that a subclass $\widehat{\Upsilon}$ is contained in both $\mathcal{C}$ and $\Omega$, but does not establish the asserted convexity of every product of two members of $\Omega$.