Generalised Erdős–Lax inequality for powers of polynomials

Let

$$P(z)=c\prod_{k=1}^{N}(z-a_k)^{ms_k},$$

where $c\in\mathbb{C}$, $m\in\mathbb{N}$, $s_k\geq 1$, $ms_k\in\mathbb{N}$, and $|a_k|\geq 1$. For any branch $f(z)=P(z)^{1/m}$, write $\|\cdot\|_{\mathbb{T}}$ for the maximum modulus on the unit circle. Generalised Erdős–Lax conjecture. One has

$$\|f'\|_{\mathbb{T}}\leq \frac{\sum_{k=1}^{N}s_k}{2}\|f\|_{\mathbb{T}},$$

with equality if and only if $|a_k|=1$ for all $k$.

This would extend the Erdős–Lax inequality to powers of polynomials without restrictions on the zeros in the exterior of the closed unit disk; the source presents it as conjectural and gives no resolution.