Bi-Ricci curvature conjecture for immersions into the unit ball

Let $X^4$ be a closed manifold and let $f:X^4\to\mathbb{B}^N$ be an immersion. Write $g=f^*g_{\mathbb{R}^N}$ for the induced metric, and let $\mathfrak{c}(f)$ denote its minimal normal curvature. Bi-Ricci curvature conjecture. If

$$\mathfrak{c}(f)<\sqrt{\frac{12}{7}},$$

then $g$ is conformally equivalent to a metric of positive bi-Ricci curvature. This would improve the available bound for $\mathcal C_N(S^2\times T^2)$; the conjecture is known when the immersion lies on the unit sphere, but extending the argument to general maps into $\mathbb{B}^N$ remains open.