q-Means and the mode of a distribution

I am not so convinced of the validity of the statement for the global maximum either. I suspect it could be true for the more restrictive case where the cumulative distribution function is convex below the mode and concave above it

Consider the following continuous density function (two triangular distributions stuck together, slightly higher on the negative side, but much wider on the positive side):

$$f(x)= \begin{cases} 0 & \text{when } x \le -2 \\ 0.1x + 0.2 &\text{when } -2 \le x \le -1\\ -0.1x &\text{when } -1 \le x \le 0\\ 0.009x &\text{when } 0 \le x \le 10\\ -0.009x+0.18 &\text{when } 10 \le x \le 20 \\ 0 & \text{when } 20 \le x \end{cases} $$

The global maximum of $f(x)$ would be $f(-1)=0.1$ compared with another local maximum of $f(10)=0.09$, but shrinkage would be towards $10$

Of course, when $q=2$ the minimising $y$ is the mean $8.9$, while when $q=1$ the minimising $y$ is the median $\sqrt{\frac{800}{9}}\approx 9.428$, so it is not particularly counter-intuitive that the minimising $y$ increases as $q$ decreases

Empirically it seems that in this example $F_q(y) \gt 1$ and $F_q(10)-1 \lt \frac12 \left(F_q(-1)-1\right)$ for all $q \in (0,1]$