thank you for asking, my proof is false and I found where. Your example seems to works. Thank you !

I would be curious if you have an idea about where i am cheating in my "proof"

Thank you for your answer ! It seems to be accurrate unfortunately for me. I will spend some time about it

Yes, the last formula is true : it is $\phi_1 \phi_2^T$ and not $\phi_2 \phi_2^T$. I edited the post to put the case $p=2$ and I think I found the general case proof. Concerning the integrals, I think they are mandatory because the matrix $\phi_1 \phi_1^T$ with $t$ fixed has rank 1 and $\int \phi_1 \phi_1^T$ can have any rank.

It means that the matrix ∫ϕ1ϕ1T - ∫ϕ2ϕ2T is positive definite. I modified the post to make it clearer, thank you.