Characterization conjecture for tensor modules over Mackey Lie algebras

Let $V$ be a countable-dimensional vector space, let $\mathfrak{g}=\mathfrak{sl}(V,V_*)$, $\mathfrak{o}(V)$, or $\mathfrak{sp}(V)$, and let $\mathfrak{g}^M$ be the corresponding Mackey Lie algebra. Let $M$ be a finite-length $\mathfrak{g}^M$-module that is integrable as a $\mathfrak{g}$-module. Let $\mathfrak{h}^M$ be a subalgebra of $\mathfrak{g}^M$. Characterization conjecture. The following conditions on $M$ are equivalent: (a) $M\in\mathbb{T}_{\mathfrak{g}^M}$; (b) $M$ is countable-dimensional; (c) $M$ is a semisimple $\mathfrak{h}^M$-module for some subalgebra $\mathfrak{h}^M\subset\mathfrak{g}^M$; (d) $M$ is a semisimple $\mathfrak{h}^M$-module for any subalgebra $\mathfrak{h}^M\subset\mathfrak{g}^M$. The conjecture would characterize tensor modules simultaneously by dimension and semisimplicity properties; the source calls it plausible and provides no resolution.