Identity crisis on tightly $C^{(2)}$-compact cardinals

Bagaria and Goldberg asked if $\kappa$ being tightly $C^{(n)}$-compact implies $\kappa \in C^{(n+1)}$ for natural numbers $n\ge 1$? Usuba gave a negative answer for $n=1$. In this paper we give a negative answer for $n=2$.