One-dimensional worst-case conjecture for gradient descent on composed operator classes

Let $\mathcal{C}_{\mu_g}^{0}$ and $\mathcal{D}_{\mu_g}^{\mu_M}$ be the operator-composed function classes defined in the source, and let $\mathcal{C}_{\mu_g}^{0,(1)}$ and $\mathcal{D}_{\mu_g}^{\mu_M,(1)}$ denote their subclasses with one-dimensional $g$ and scalar linear operator $M$. One-dimensional worst-case conjecture. The worst-case performances are attained in these one-dimensional subclasses:

$$w(\mathcal{C}_{\mu_g}^{0};h)=w(\mathcal{C}_{\mu_g}^{0,(1)};h),\qquad w(\mathcal{D}_{\mu_g}^{\mu_M};h)=w(\mathcal{D}_{\mu_g}^{\mu_M,(1)};h).$$

The conjecture is motivated by the authors' numerical experiments, but the supplied text gives no proof or resolution.