The absence of unbounded d-weakly convergent sequences in normed linear spaces

Let $(X,\lVert\cdot\rVert)$ be a normed linear space, and let $d$ be the metric induced by $\lVert\cdot\rVert$. A sequence in $X$ is unbounded if it is not bounded with respect to this norm, and it converges $d$-weakly according to the metric-functional definition: for every metric functional $\operatorname{\mathbf{h}}$ on $X$, its limit $z$ satisfies

$$\liminf_{n\to\infty}\operatorname{\mathbf{h}}(x_n)\geq \operatorname{\mathbf{h}}(z).$$

The absence conjecture. There is no normed linear space where an unbounded sequence converges $d$-weakly.

The paper proves this absence for $\ell_1$, $C[0,1]$, and normed linear spaces whose dual is strictly convex, while the assertion for arbitrary normed linear spaces is left unresolved.