A small Banach space $C(K)$ without nice renormings

We prove that consistently ${\omega }_1<\mathfrak{c}$ and there exists a compact space $K$ whose Banach space $C(K)$ of continuous real-valued functions is Grothendieck, has density ${\omega }_1$, and admits no renorming which is strictly convex or sequentially Kadets–Klee.