BTP metrics are BAS under homogeneous universal-cover hypotheses

Let $(M^n,g)$ be a compact Hermitian manifold whose universal cover is a homogeneous Hermitian manifold without Kähler de Rham factor. A Hermitian metric is BTP if its Bismut connection has parallel torsion, and it is BAS if it is naturally reductive with respect to a transitive action of isometric biholomorphisms. Assume that $g$ is BTP and that one of the following holds:

1. $M^n$ has finite fundamental group.
2. The universal cover of $(M^n,g)$ is a Lie group equipped with a left-invariant metric and a compatible left-invariant complex structure.

BTP-to-BAS conjecture. Under these assumptions, $g$ must be BAS.

The conjecture is motivated by the preceding example of a compact, non-simply connected homogeneous $4$-fold carrying a BTP metric that is not BAS, and asks whether the stated restrictions on the fundamental group or universal cover force natural reductivity. Its resolution is not given in the supplied text.