My former doctoral student Lubomyr Zdomskyy has resolved this problem, noticing that adding $\omega_2$ Cohen reals to a model of GCH produces a model in which the cardinal $\omega_1$ admits a monotone cofinal map $\omega^\omega\to(\omega_1)^{\omega_1}$.

Alternatively the same result can be derived from the existence of a K-Lusin set of cardinality $\mathfrak c=2^{\omega_1}$ in $\omega^\omega$.

A subset $L$ of a Polish space $X$ is called a K-Lusin set in $X$ if for every compact subset $K\subset X$ the intersection $K\cap L$ is countable.

Theorem (Zdomskyy). If there exists a $K$-Lusin set $L\subset\omega^\omega$ of cardinality $|L|=\mathfrak c=2^{\omega_1}$, then there exists a mootone cofinal map $f:\omega^\omega\to(\omega_1)^{\omega_1}$.

Proof. Let $g:L\to (\omega_1)^{\omega_1}$ be any surjective map. Observe that for every $x\in\omega^\omega$ the set $L\cap {\downarrow}x=\{y\in L:y\le x\}$ is countable. Then the formula $f(x)=\sup(g(L\cap{\downarrow}x)$ defines a required cofinal monotone map $f:\omega^\omega\to(\omega_1)^{\omega_1}$. $\square$

The existence of a K-Lusin set of cardinality $\mathfrak c$ in the Cohen model was established by Bartoszynski and Halbeisen.

My former doctoral student Lubomyr Zdomskyy has resolved this problem, noticing that adding $\omega_2$ Cohen reals to a model of GCH produces a model in which the cardinal $\omega_1$ admits a monotone cofinal map $\omega^\omega\to(\omega_1)^{\omega_1}$.

Alternatively the same result can be derived from the existence of a K-Lusin set of cardinality $\mathfrak c=2^{\omega_1}$ in $\omega^\omega$.

A subset $L$ of a Polish space $X$ is called a K-Lusin set in $X$ if for every compact subset $K\subset X$ the intersection $K\cap L$ is countable.

Theorem (Zdomskyy). If there exists a $K$-Lusin set $L\subset\omega^\omega$ of cardinality $|L|=\mathfrak c=2^{\omega_1}$, then there exists a mootone cofinal map $f:\omega^\omega\to(\omega_1)^{\omega_1}$.

Proof. Let $g:L\to (\omega_1)^{\omega_1}$ be any surjective map. Observe that for every $x\in\omega^\omega$ the set $L_x=\{y\in L:y\le x\}$ is countable. Then the formula $f(x)=\sup(g(L_x))$ defines a required cofinal monotone map $f:\omega^\omega\to(\omega_1)^{\omega_1}$. $\square$

The existence of a K-Lusin set of cardinality $\mathfrak c$ in the Cohen model was established by Bartoszynski and Halbeisen.