Evidence for $\mathsf{P} \neq \mathsf{PP}$ if the polynomial hierarchy collapses? We think that $\mathsf{PH}$ does not collapse, and that $\mathsf{PP}$ is not in $\mathsf{P}$.

Suppose on the contrary that $\mathsf{PH}$ does collapse, say even $\mathsf{P}= \mathsf{NP}$.

$\mathsf{PP}$ still seems like it would be of reach. 

What evidence do we have that $\mathsf{P} \neq \mathsf{PP}$, under that assumption, i.e., evidence that does not apply to $\mathsf{P}$ vs. $\mathsf{NP}$?

$\underline{\mbox{Assumption}}$: Note if $\mathsf{SAT}$ is in linear time then perhaps $\mathsf{P=PSPACE}$ might follow as composition of linear time is linear time? 

$\underline{\mbox{Following is only based on if above assumption follows}}$:


Perhaps the problem would be 

1a. why believe $\mathsf{SAT}$ is in superlinear time

1b. and if there is reason to believe $\mathsf{SAT}$ is in superlinear time then why believe $\mathsf{PP}$ requires $\Omega(\log\log n)$ quantifications?
Do evidences provided by https://eccc.weizmann.ac.il/report/2018/107/ and http://arxiv.org/abs/1504.03398 still hold up in face of $\mathsf{P=NP}$ and $\mathsf{SAT}$ in linear time ($\mathsf{P=NP}$ seems to invalidate second)?
The scaled down version of $\mathsf{PH}$ versus $\mathsf{PP}$ is $\mathsf{AC}^0$ versus $MAJ \circ \mathsf{AC}^0$, and we know that for the latter there is an exponential separation. Of course, this separation doesn't propagate exponentially up, but you could take this as philosophical evidence that $\mathsf{PH}$ is different enough from $\mathsf{PP}$ that perhaps that remains true even if $\mathsf{P} = \mathsf{NP} = \mathsf{PH}$. (In fact, following this line of reasoning, I'd guess you can build an oracle relative to which $\mathsf{P} = \mathsf{NP} \neq \mathsf{PP}$.)