Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$

Question

Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$

I would like to prove (or disprove) that \begin{align*} \operatorname{Re}\big(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi)\big)\geq0\qquad\text{for all $\xi\in \Bbb R^d$} \end{align*} Note that for $\tilde{u}(x)=u(-x)$, \begin{align*} \operatorname{Re}\big(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi)\big)= \operatorname{Re}\big( \widehat{\tilde{u}* F\circ u}(\xi)\big)= \widehat{\tilde{u}* F\circ u}(\xi)+ \widehat{F\circ \tilde{u}* u}(\xi). \end{align*}

PS. This problem is suggested by the decay of generalized porous medium equation $$\partial_t u= \Delta F\circ u.$$ In the Fourier variables we expect the time decay $|\widehat{u(t)}|^2\leq c |\widehat{u(0)}|^2$. The latter is true if we have $$\partial_t |\widehat{u}(\xi)|^2= -2|\xi|^2\operatorname{Re}\big(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi)\big)\leq 0.$$

My main problem can be resolved if one shows that $|\widehat{u(t)}|^2\leq c |\widehat{u(0)}|^2,\, c>0$.

Any idea, partial answer or reference it warmly appreciated.