Ricci eigenvalue conjecture for immersions into the unit ball

Let $X^n$ be a closed manifold and let $f:X^n\to\mathbb{B}^N$ be an immersion. Write $g=f^*g_{\mathbb{R}^N}$ for the induced metric, and let $\mathfrak{c}(f)$ denote its minimal normal curvature. Suppose that $\tilde g$ is a metric conformally equivalent to $g$, and let $\tilde\lambda_1\leq\dots\leq\tilde\lambda_4$ be the eigenvalues of the Ricci tensor $\widetilde{\operatorname{Ric}}$ of $\tilde g$ at any point. Ricci eigenvalue conjecture. If

$$\mathfrak{c}(f)<\sqrt{\frac{9}{5}},$$

then $g$ is conformally equivalent to a metric $\tilde g$ such that

$$\tilde\lambda_4<\tilde\lambda_1+\tilde\lambda_2+\tilde\lambda_3.$$

This condition could yield a matching lower bound for $\mathcal C_N(S^2\times T^2)$, but the conjecture remains open.