Lu's conjecture for minimal surfaces in codimension two

Let $M^2\to\Sph^4$ be a closed minimal immersion, let $S$ be the squared norm of its second fundamental form, and let $\lambda_1\geq\lambda_2\geq0$ be the eigenvalues of Lu's fundamental matrix. We prove that if $S+\lambda_2$ is constant and larger than $2$, then $S+\lambda_2\geq3$. Thus Lu's second-gap conjecture holds for minimal surfaces in codimension two.
Together with the hypersurface result of Peng--Terng and the counterexamples of Li--Zhao in every codimension $m\geq3$, our theorem completes the codimension picture of Lu's second-gap conjecture for minimal surfaces: it holds precisely for $m=1,2$ and fails for every $m\geq3$.