The conjecture that the entropy sum lower-bound constant can be improved to one-half

Let $G={\mathbf{F}}_p$ or $G=\mathbf{R}$, let $0<\epsilon<1/2$, and let $X$ and $Y$ be independent discrete random variables taking values in $G$, under the hypotheses of Theorem 1. The theorem currently gives

$$\operatorname{\mathbf{H}}(X+Y)\geq \frac{1}{2}\operatorname{\mathbf{H}}(X)+\frac{1}{2}\operatorname{\mathbf{H}}(Y)+\frac{1}{8}-\epsilon.$$

Entropy sum lower-bound conjecture. The constant $1/8$ can be improved to $1/2$, meaning that

$$\operatorname{\mathbf{H}}(X+Y)\geq \frac{1}{2}\operatorname{\mathbf{H}}(X)+\frac{1}{2}\operatorname{\mathbf{H}}(Y)+\frac{1}{2}-\epsilon.$$

The value $1/8$ is known for the stated non-identically distributed setting, whereas the value $1/2$ holds when the variables are identically distributed and is suggested by Shannon's continuous entropy power inequality. Improving the constant in the general independent setting remains open.