The conjectured lower bound for Riesz means of the Möbius function

Let $M_\tau(x)$ be the Riesz mean

$$M_\tau(x):=\frac{1}{\Gamma(\tau+1)}\sum_{n\leq x}\mu(n)\left(1-\frac{n}{x}\right)^\tau,$$

where $\mu$ is the Möbius function. Let $\rho$ be any non-trivial zero of the Riemann zeta-function, with multiplicity $m(\rho)$, and let $\tau=\tau(x)$ be any monotone positive-valued function. The conjecture.

$$M_\tau(x)=\Omega\left(x^{1/2}(\log x)^{m(\rho)-1}(\tau/e)^{-\tau-1}\right).$$

The authors motivate this by expecting the contribution from a multiple zeta zero to dominate the behavior of $M_\tau(x)$, but the source gives no resolution evidence.