Let $s^2=\frac{1}{n}\sum_1^n( X_i-\bar{X})^2$ and $\widetilde{X}$ denote the sample median. Is $s^2\leq \widetilde{X}(1-\widetilde{X})$ true

Question

$X_i\in[0,1]$

Let $s^2=\frac{1}{n}\sum_1^n( X_i-\bar{X})^2$ and $\widetilde{X}$ denote the sample median.

Is the following true?

$$s^2\leq \widetilde{X}(1-\widetilde{X})$$

I couldn't find any way to relate the above two (other than specific cases). For example, if I take $10\ X_i$'s to be zero, and $10$ one, and $\widetilde{X} = .5$ then equality occurs. But other than this case, I can't find any particular way to get a general result.