Let $A \in {\Bbb C}^{4 \times 4}$ be a traceless matrix with Frobenius norm smaller than $1$. Let $\lambda_i$ be $i$-th largest (by modulus) eigenvalue of matrix $A$. Let $\sigma_i$ be $i$-th largest singular value of matrix $A$. Is the following statement true?
$$ \lvert \lambda_1 \rvert \leq \sqrt{\sigma_2^2 + \sigma_3^2 + \sigma_4^2} $$
I am unable to prove the statement, yet I was unable to find a counterexample. I am aware that I must use Weyl's inequalities (or its consequences), but I cannot find a way.