Implication from strong convergence of (η^(k), b^(k)) to strong convergence of amalgamated free copies

Determine whether the following implication holds: if the pairs (η^(k), b^(k)) converge strongly in covariance law to (η, b), then the pairs (b^(k), S^(k)) converge strongly (i.e., operator norms of all noncommutative polynomials converge) to (b, S), where S^(k) is a tuple of freely independent copies, with amalgamation over M_{n(k)}, of the (M_{n(k)}, η^(k))-semicircular family X_free^(k), and S is a tuple of freely independent copies, with amalgamation over B, of the (B, η)-semicircular family X.

Background

The paper establishes strong convergence of Gaussian and deterministic matrix models to operator-valued semicircular families under an assumption that involves convergence of amalgamated free copies S^k of the (M_{n(k)}, η^k)-semicircular family together with b^k. The authors note this is a natural hypothesis for their proof strategy, which compares Gaussian matrices to their operator-valued semicircular counterparts.

They remark that it would be more natural to assume strong convergence in covariance law of (η^k, b^k) to (η, b). However, it is unknown in general whether this stronger-looking assumption implies the required strong convergence of the pairs (b^k, S^k) to (b, S). For fixed base algebras, strong convergence is known to pass to free products (Skoufranis; Pisier), but the present setting involves varying base algebras and operator-valued covariance, for which this implication is not established.