The square-root commutator conjecture for positive contractions

Let $H$ and $A$ belong to a $C^{*}$-algebra, with $H$ a positive contraction satisfying $0\leq H\leq1$ and $\left\Vert A\right\Vert\leq1$. For the square-root function $f(x)=\sqrt{x}$, let $\gamma_f$ denote the corresponding commutator modulus for positive contractions. Square-root commutator conjecture. One has

$$\gamma_f(\delta)=\sqrt{\delta}.$$

Equivalently,

$$\left\Vert \left[H^{\frac{1}{2}},A\right]\right\Vert\leq\left\Vert [H,A]\right\Vert^{\frac{1}{2}}.$$

This asserts the sharp square-root modulus of continuity for commutators under functional calculus of positive contractions. The supplied context identifies the assertion as something that was not clearly attributed and gives no evidence of resolution.