Namba forcing is stationary-preserving and forces $cf(\omega_2^{\mathbf{V}}) = \omega$. Ronald Jensen used $\mathcal{L}$-forcing to iterate Namba posets in order to solve the extended Namba problem: for any strongly inaccessible $\kappa$, he constructed a stationary-preserving forcing notion $\mathbb{P}_{\kappa}$ that forces all ground model regular cardinals in the interval $(\omega_1, \kappa)$ to have cofinality $\omega$, while preserving the cofinality of $\kappa$. His construction allows $\mathbb{P}_{\kappa}$ to add no reals.

To formalise my question, let me use $Nb(\alpha, \beta)$ to denote the statement

$``$there is a stationary-preserving forcing notion $\mathbb{P}$ such that

- for all $\lambda \in \mathbf{V} \cap (\alpha, \beta)$, if $\lambda$ is regular, then $\Vdash_{\mathbb{P}} cf(\lambda) = \omega$, and
- $\Vdash_{\mathbb{P}} cf(\beta) > \omega$.$"$

In solving the extended Namba problem in the affirmative, Jensen showed that $Nb(\omega_1, \beta)$ is true when $\beta$ is strongly inaccessible. In another example of $\mathcal{L}$-forcing he gave, he used a modified construction to get $Nb(\omega_1, \beta^+)$ for any cardinal $\beta$ of cofinality $\omega_1$. The forcing notions he presented have the additional property of not adding reals, which I do not require.

My question is thus, is it known if $Nb(\omega_1, \beta)$ is true for all regular cardinals $\beta$?