Does CMC(ℵ0,∞) imply regularity of ℵ1?

Determine whether the Countable Multiple Choice principle CMC(ℵ0,∞) implies that ℵ1 is regular, i.e., whether CMC(ℵ0,∞) implies cf(ℵ1)=ℵ1 in ZF.

Background

Theorem s5:t6 relates P-space properties of S(κ,[κ]^{≤ω}) and 2^κ[[κ]^{≤ω}] to weak choice principles for uncountable well-ordered κ. The authors note that while some directions are proved under additional hypotheses, the implication from the Countable Multiple Choice CMC(ℵ0,∞) to the regularity of ℵ1 remains unresolved.