Cardinality and Ramsey-ultrafilter conjecture for Rudin–Keisler intervals

Let \boldsymbol\beta\boldsymbol\thinspace\boldsymbol\backslash\boldsymbol\beta\boldsymbol\thinspace\boldsymbol\text{?} be the space of ultrafilters on \boldsymbol\beta\boldsymbol\thinspace\boldsymbol\text{?}, let $\boldsymbol u$ be an ultrafilter, and let $[\boldsymbol u,\boldsymbol u^{\tensor 2}]$ denote the interval of isomorphism classes between $\boldsymbol u$ and its tensor square in the Rudin–Keisler order. Write $\beth_1$ for the cardinality of the continuum.

Cardinality and Ramsey-ultrafilter conjecture. For every ultrafilter $\boldsymbol u$,

$$|[\boldsymbol u,\boldsymbol u^{\tensor 2}]|\in\omega\cup\{\beth_1\}.$$

Moreover, the following are equivalent: there exists a Ramsey ultrafilter; for every \boldsymbol\boldsymbol\text{kappa}\in\omega\cup\{\beth_1\}, there is an ultrafilter $\boldsymbol u$ such that

|[\boldsymbol u,\boldsymbol u^{\tensor 2}]|=\boldsymbol\boldsymbol\text{kappa}.

The accompanying hint indicates that finite lower Rudin–Keisler intervals yield finite intervals here, while an infinite lower interval produces $\beth_1$ pairwise non-isomorphic intermediate ultrafilters. The source presents this as a question, so its resolution is not established in the supplied text.