Googleology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to each countable limit ordinal:

It is an open problem whether an ordinal hierarchy can be defined without using fundamental sequences. More explicitly, the problem concerns whether there is a model of $ZF$ such that there exists an $F : \omega_1 \rightarrow (\mathbb{N} \rightarrow \mathbb{N})$ where for all $\alpha > \beta, F(\alpha)$ eventually outgrows $F(\beta)$, but there does not exist an $S: \omega_1 \cap \text{Lim} \rightarrow (\mathbb{N} \rightarrow \omega_1)$ such that for all $\alpha, \sup(R) = \alpha$ where $R$ is the range of $S(\alpha)$.

My question is, is it true that the question of whether there exists such a model of $ZF$ is an open problem?

Googology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to each countable limit ordinal:

It is an open problem whether an ordinal hierarchy can be defined without using fundamental sequences. More explicitly, the problem concerns whether there is a model of $ZF$ such that there exists an $F : \omega_1 \rightarrow (\mathbb{N} \rightarrow \mathbb{N})$ where for all $\alpha > \beta, F(\alpha)$ eventually outgrows $F(\beta)$, but there does not exist an $S: \omega_1 \cap \text{Lim} \rightarrow (\mathbb{N} \rightarrow \omega_1)$ such that for all $\alpha, \sup(R) = \alpha$ where $R$ is the range of $S(\alpha)$.

My question is, is it true that the question of whether there exists such a model of $ZF$ is an open problem?

Googleology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to each countable limit ordinal:

It is an open problem whether an ordinal hierarchy can be defined without using fundamental sequences. More explicitly, the problem concerns whether there is a model of $ZF$ such that there exists an $F : \omega_1 \rightarrow (\mathbb{N} \rightarrow \mathbb{N})$ where for all $\alpha > \beta, F(\alpha)$ eventually outgrows $F(\beta)$, but there does not exist an $S: \omega_1 \cap \text{Lim} \rightarrow (\mathbb{N} \rightarrow \omega_1)$ such that for all $\alpha), \sup(R) = \alpha$ where $R$ is the range of $S(\alpha)$.

My question is, is it true that the question of whether there exists such a model of $ZF$ is an open problem?

Googleology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to each countable limit ordinal:

It is an open problem whether an ordinal hierarchy can be defined without using fundamental sequences. More explicitly, the problem concerns whether there is a model of $ZF$ such that there exists an $F : \omega_1 \rightarrow (\mathbb{N} \rightarrow \mathbb{N})$ where for all $\alpha > \beta, F(\alpha)$ eventually outgrows $F(\beta)$, but there does not exist an $S: \omega_1 \cap \text{Lim} \rightarrow (\mathbb{N} \rightarrow \omega_1)$ such that for all $\alpha, \sup(R) = \alpha$ where $R$ is the range of $S(\alpha)$.

My question is, is it true that the question of whether there exists such a model of $ZF$ is an open problem?