A sharp pinching theorem for closed minimal Legendrian submanifolds

Let $F:M^n\to {\mathbb{S}}^{2n+1}$ be a minimal Legendrian immersion of a connected closed manifold, where $n\ge 3$, and let $B$ be its second fundamental form. We prove that if $\mathrm{\mid }B{\mathrm{\mid }}^2\le (n-1)(n+2)\mathrm{/}n$, then, up to a unitary transformation, $F$ parametrizes a totally geodesic sphere or is a finite covering of the image of the minimal Calabi product of a point and a totally geodesic Legendrian sphere. This proves Conjecture 1 of Luo and Sun [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), p. 548].