Spectrum invariance under toral and spherical Aluthge transforms of commuting 2-variable weighted shifts

Let $W_{(\alpha,\beta)}$ be a commuting $2$-variable weighted shift, and suppose that its toral and spherical Aluthge transforms, denoted by $\widetilde{W}_{(\alpha,\beta)}$ and $\widehat{W}_{(\alpha,\beta)}$, are also commuting. Spectrum-invariance conjecture. The three operators $W_{(\alpha,\beta)}$, $\widetilde{W}_{(\alpha,\beta)}$, and $\widehat{W}_{(\alpha,\beta)}$ should have the same Taylor spectrum and the same Taylor essential spectrum. The claim extends the preceding results on spectral invariance for these transforms and concerns the relationship between the Taylor spectra and Taylor essential spectra of commuting multivariable weighted shifts.