$\sum\limits_i a_i^2\sum\limits_i b_i^2+\left(\sum\limits_ia_i b_i\right)^2\geq \sqrt{\sum\limits_i a_i^4\sum\limits_i b_i^4}+\sum\limits_ia_i^2b_i^2$

Question

I have no idea about how to prove (or disprove) the following inequality: $$ \left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right)+\left(\sum_{i=1}^na_i b_i\right)^2\geq \sqrt{\left(\sum_{i=1}^n a_i^4\right)\left(\sum_{i=1}^n b_i^4\right)}+\sum_{i=1}^na_i^2b_i^2,\quad a_i,b_i\in\mathbb{R}, \ n>1. $$ I ran some numerical simulations and no counterexample showed up yet.

Note 1. The inequality holds true for $n=2$, as shown here.

Note 2. This conjecture was formulated by Fedor Petrov in an attempt to provide a solution to a particular case of this question.

---

EDIT. The inequality has been finally proved here.