Strongly regular Banach spaces with big weakly open subsets in the unit ball

We construct, given $1<p<\infty$, a Banach space $Y$ and a closed, convex and symmetric set $L\subseteq B_Y$ with the following properties:
1) $Y^{**}$ is strongly regular (henceforth, $Y$ is strongly regular).
2) Every non-empty relatively weakly-star open subset of $\overline{L}^{w^*}$ (the $w^*$ closure of $L$ in $Y^{**}$) has radius one. In particular, every non-empty relatively weakly open subset of $L$ has radius $1$.
3) Every non-empty relatively weakly open subset of $L$ has diameter, at least, $2^\frac{1}{p}$.
This constitutes an advance to the question whether there exists a strongly regular Banach spaces satisfying that every non-empty relatively weakly open subset of the unit ball has radius $1$. As a partial answer, we get that for every $\varepsilon>0$ there exists a strongly regular Banach spaces where weakly open subsets have radius, at least, $1-\varepsilon$.