Fractional-power numerical-radius conjecture for accretive matrices

Let $A\in\Gamma_n$, where $\Gamma_n$ denotes the class of accretive $n\times n$ matrices, and let $t$ satisfy $0<t<1$. Fractional-power numerical-radius conjecture.

$$\omega(A^t)\geq \omega^t(A).$$

This conjecture proposes that the numerical-radius inequality for natural powers reverses for every fractional power in $(0,1)$; the source provides simulations and proves the claim when $t$ is the reciprocal of a natural number, but leaves the general fractional case open.