Does a strictly convex and weak metrizable unit sphere of a Banach space imply locally uniform convexity?

Question

I'm trying to find a proof for this question

Let $X$ be a Banach space with strictly convex norm and a metrizable unit sphere, $S_X=\{x\in X: \|x\|=1\}$, for the weak topology.

Does $X$ admits an equivalent locally uniformly convex norm? recall that

1. A Banach space $(X, \|\cdot\|)$ is said to be locally uniformly convex or locally uniformly rotund (LUR) if for all $x, x_n \in X$ satisfying $$\lim_{n\to \infty}(2\|x\|^2+2\|x_n\|^2-\|x+x_n\|^2)=0$$ we have $\displaystyle\lim_{n\to \infty}\|x_n-x\|=0.$

2. A Banach space $(X, \|\cdot\|)$ is said to be strictly convex or rotund (R) if for all $y, x \in X$ satisfying $$(2\|y\|^2+2\|x\|^2-\|y+x\|^2)=0$$ we have $x=y.$

Any help will be appreciated! Thanks