Cofibrancy of K ⊗ O as a right O-module for general operads

Determine whether, for a reduced operad O in the category of symmetric spectra and a based simplicial set K, the O-bimodule K ⊗ O defined levelwise by (K ⊗ O)(n) = K^∧n ∧ O(n) is cofibrant as a right O-module when O ≠ Com, in the model structure on right O-modules induced from the positive symmetric spectra model structure.

Background

In Sections 4–5 the paper proves that for the reduced commutative operad Com, the bimodule K ⊗ Com is cofibrant as a right Com-module. This cofibrancy is pivotal for identifying K ⊗ I with (K ⊗ Com) ∘_Com I and for constructing canonical filtrations with explicitly described cofibers.

Section 9 extends these constructions to a general reduced operad O, showing that K ⊗ I ≅ (K ⊗ O) ∘_O I. However, the specific argument establishing cofibrancy of K ⊗ Com does not generalize immediately to arbitrary operads, leaving open whether K ⊗ O is cofibrant as a right O-module in general. Establishing cofibrancy would enable the same homotopical control and filtration properties for O-algebras that are available in the Com case.