Conformal-class characterization of negatively infinite Yamabe constant

Let $M$ be a nonempty manifold of dimension $n\geq 3$, and let $g\in\operatorname{Metr}(M)$. For a metric $\overline{g}$ conformal to $g$, write $({\operatorname{scal}}_{\overline{g}})_-$ for the negative part of its scalar curvature and measure its $L^{n/2}$-norm with respect to $\overline{g}$. Conformal-class characterization. The equality $Y_M(g)=-\infty$ holds if and only if

$$\big\lVert({\operatorname{scal}}_{\overline{g}})_-\big\rVert_{L^{n/2}(\overline{g})}=\infty$$

for every metric $\overline{g}$ in the conformal class of $g$. The preceding discussion explains that finite $L^{n/2}$-norm implies $Y_M(g)>-\infty$, while the converse for the original metric can fail because the norm is not conformally invariant; the statement proposes that this is the only obstruction.