Koumandos–Ruscheweyh weaker conjecture for partial sums

Let $s_n^{\mu}(z)=\displaystyle\sum_{k=0}^n \frac{(\mu)_k}{k!}z^k$ for $0<\mu\leq1$, and let $\rho\in(0,1]$. Let $\mu^{\ast}(\rho)$ be the number defined by

$$\int_{0}^{(\rho+1)\pi}\frac{\sin(t-\rho\pi)}{t^{1-\mu}}dt=0.$$

Koumandos–Ruscheweyh weaker conjecture. For every $z\in\mathbb{D}$, $n\in\mathbb{N}$, and $0<\mu\leq\mu^{\ast}(\rho)$,

$$\operatorname{Re}\left((1-z)^{2\rho-1}s_n^{\mu}(z)\right)>0.$$

Moreover, $\mu^{\ast}(\rho)$ is the largest number with this property. This is presented as a weaker form of the preceding subordination conjecture, and the supplied text gives no resolution status.