$$4s_5 s_1^3 + s_2^4 \geq 5s_4 s_2 s_1^2.$$
Using the AM-GM inequality, $4s_5 s_1^3 + s_2^4 \geq 5 [s_5 s_1^{12} s_2^4]^{1/5}$, so it suffices to prove that
$$s_5^4 s_2^4 s_1^{12} \geq s_4^5 s_2^5 s_1^{10},$$
and thus, using the positivity of $s_k$, that $s_5 s_1^2 \geq s_4 s_2$.

Observe that $s_5 s_3 \geq s_4^2$, since, on expansion $\sum_{i=1}^n a_i^8$ occurs on both sides, and each term $(a_i^5 a_j^3 + a_i^3 a_j^5) = (a_i^3 a_j^3)(a_i^2 + a_j^2) \geq a_i^3 a_j^3 (2a_i a_j) = 2a_i^4 a_j^4$ for all $i < j$.

Observe also that $s_5 s_1 \geq s_3^2$, since, on expansion, $\sum_{i=1}^n a_i^8$ occurs on both sides, and each term $(a_i^5 a_j + a_i a_j^5) = a_i a_j (a_i^4 + a_j^4) \geq 2a_i a_j (a_i^2 a_j^2) = 2a_i^3 a_j^3$, $(i < j)$.

Observe further that $s_5 s_1 \geq s_4 s_2$, since
$$(a_i^5 a_j + a_i a_j^5) - (a_i^4 a_j^2 + a_i^2 a_j^4) = a_i a_j (a_i - a_j)^2 (a_i^2 + a_i a_j + a_j^2) \geq 0.$$
Now $s_5 s_3^2 s_1^2 \geq s_2^2 s_4 s_1^2 \geq s_5 s_4 s_3^2 s_1 \geq s_4^5 s_2^2 s_2$, and the result follows.