Singular Values & the SVD · Challenge problem

A real $4\times 4$ matrix $A$ has $\|A\|_2 = 5$ and $\|A\|_F = 6$. Find the largest possible value of its nuclear norm $\|A\|_* = \sigma_1 + \sigma_2 + \sigma_3 + \sigma_4$. Give your answer in simplest radical form.