The measurable repeat-point consistency conjecture for mutual stationarity

Let $\langle \kappa_n \mid n < \omega\rangle$ be a sequence of infinitely many measurable cardinals. For each $n$, suppose that $\kappa_n$ carries an $(\omega,\kappa_{n-1}^+)$ repeat point measure, with the convention that $\kappa_{n-1}^+$ is replaced by $\aleph_0$ when $n=0$. A sequence of stationary sets $S_n \subset \omega_{2n+1}$ has common fixed cofinality when all the sets $S_n$ consist of ordinals of the same fixed cofinality. Measurable repeat-point consistency conjecture. The assertion that every sequence of stationary sets $S_n \subset \omega_{2n+1}$ of common fixed cofinality is consistent relative to such a sequence of measurable cardinals. The conjecture proposes that the supercompact-level consistency assumption previously established can be lowered to one involving measurable cardinals with repeat point measures; the source gives no resolution of this expected strengthening.