Monotonicity conjecture for c-convolution

Let $\mu$ and $\nu$ be compactly supported probability measures, let $c\in[0,\infty]$, and let $\mu\oplus_c\nu$ denote their $c$-convolution. Let $f\in C^{4}(\mathbb{R})$ satisfy $f^{(4)}\geq 0$. Monotonicity conjecture for c-convolution. The function

$$c\longmapsto\int f(t)\,\operatorname{d}(\mu\oplus_c\nu)(t)$$

is decreasing in $c$. The $c$-convolution interpolates formally between classical convolution at $c=0$ and free convolution at $c=\infty$, but it is not known whether $\mu\oplus_c\nu$ is always a probability measure; consequently, the conjecture concerns a proposed generalization of the comparison results in the paper.