The global-dimension conjecture for sufficiently nice localizations

Let $R$ be a ring spectrum, let $v\in \pi_*R$ be a homogeneous element, and consider the localization map

$$R\to R[v^{-1}].$$

Here $R[v^{-1}]$ denotes the localization obtained by inverting $v$, and a localization is called sufficiently nice when it satisfies the additional hypotheses intended in the source. Global-dimension conjecture. If the localization $R\to R[v^{-1}]$ is sufficiently nice, then it cannot reduce global dimension by more than $1$; equivalently,

$$\operatorname{gl.dim}(R[v^{-1}])\geq \operatorname{gl.dim}(R)-1.$$

This conjecture arose from the problem of determining the global dimensions of the real and complex K-theory spectra. The source does not specify here what “sufficiently nice” means or report a resolution of the conjecture.