Strictness conjecture for inclusions of BPF and nef b-divisor spaces

Let $X$ be a projective manifold of dimension at least $2$, and let $\mathcal{X}$ be its Riemann–Zariski space. Consider the continuous injections

$$\operatorname{N}_{\operatorname{BPF}}^{1}(\mathcal{X})\hookrightarrow \operatorname{Vect}(\operatorname{Nef^1}(\mathcal{X}))\hookrightarrow \operatorname{N^1_{\Sigma}}(\mathcal{X})\hookrightarrow \operatorname{N}^1_{\omega}(\mathcal{X})=\operatorname{N}^{1,\vee}_{\operatorname{BPF}}(\mathcal{X})\hookrightarrow \operatorname{w-N}^1(\mathcal{X}).$$

Strictness conjecture. All the inclusions in this sequence are strict.

These inclusions compare several Banach and completed spaces of $b$-divisor classes used in the study of degree growth of rational self-maps. The first and last injections were previously known, while the equality and the two intermediate injections are established in the source; the strictness assertion remains open.