Equality of left and right weak Gorenstein global dimensions

Let $R$ be a ring. Its left and right weak Gorenstein global dimensions are defined by

$$l\operatorname{.Gwdim}(R)={\rm \sup}\{\operatorname{Gfd}_R(M)\mid M\text{ is a left }R\text{-module}\}$$

and

$$r\operatorname{.Gwdim}(R)={\rm \sup}\{\operatorname{Gfd}_R(M)\mid M\text{ is a right }R\text{-module}\}.$$

Equality conjecture. For any ring $R$,

$$l\operatorname{.Gwdim}(R)=r\operatorname{.Gwdim}(R).$$

The equality is known when $R$ has finite weak global dimension and when $R$ is left and right Noetherian. The conjecture asks whether it holds for arbitrary rings.