When is $\Vert \Vert f \Vert_p \Vert_q \leq \Vert\Vert f\Vert_q\Vert_p$? Under which condition on two positive real numbers $p,q$ (not necessarily Hölder-conjugated) holds the following inequality for functions where the integral exists?
$$\left(\int_{\Omega_1} \left( \int_{\Omega_2} \vert f(s,t)\vert^p d\mu(s)\right)^{\frac{q}{p}} d\nu(t)\right)^{\frac{1}{q}} \leq
\left(\int_{\Omega_2} \left( \int_{\Omega_1} \vert f(s,t)\vert^q d\nu(t)\right)^{\frac{p}{q}} d\mu(s)\right)^{\frac{1}{p}}$$
For $p=q$ this is Fubini. Furthermore, if $f(s,t)=g(s)h(t)$ we also have equality. My conjecture is that this holds for $p\leq q$ but I have not found a way to prove it yet. I have in particular tried the case $p=1$ and then applying Hölder's inequality but I did not reach any conclusion. Any hint on how to get started is greatly appreciated.

Assuming $p\le q$ and letting $g(s,t)=|f(s,t)|^p$ and $\frac1r+\frac pq=1$, this is equivalent to
$$
\begin{align}
\left(\int_{\Omega_1}\left(\int_{\Omega_2}g(s,t)\,\mathrm{d}\mu(s)\right)^{q/p}\,\mathrm{d}\nu(t)\right)^{p/q}
&=\sup_{\|h\|_r=1}\int_{\Omega_1}\int_{\Omega_2}g(s,t)\,\mathrm{d}\mu(s)\,h(t)\,\mathrm{d}\nu(t)\tag1\\
&=\sup_{\|h\|_r=1}\int_{\Omega_2}\int_{\Omega_1}g(s,t)\,h(t)\,\mathrm{d}\nu(t)\,\mathrm{d}\mu(s)\tag2\\
&\le\int_{\Omega_2}\sup_{\|h\|_r=1}\int_{\Omega_1}g(s,t)\,h(t)\,\mathrm{d}\nu(t)\,\mathrm{d}\mu(s)\tag3\\
&=\int_{\Omega_2}\left(\int_{\Omega_1}g(s,t)^{q/p}\,\mathrm{d}\nu(t)\right)^{p/q}\,\mathrm{d}\mu(s)\tag4
\end{align}
$$
Explanation:
$(1)$: Converse Hölder Inequality
$(2)$: Fubini's Theorem
$(3)$: an $h$ for each $s$ is at least as big as one $h$ for all $s$
$(4)$: Hölder's Inequality
