Perdomo's index conjecture for minimal hypersurfaces in spheres

Let $x:M^n\rightarrow\mathbb{S}^{n+1}$ be a full, closed, minimal hypersurface in $\mathbb{S}^{n+1}$, with $n\geq 3$. Here $\operatorname{Ind}(M)$ denotes the Morse index of $M$. Perdomo's conjecture. One has

$$\operatorname{Ind}(M)\geq n+3,$$

with equality if and only if $x(M)$ is congruent to one of the Clifford minimal hypersurfaces

$$\mathbb{S}^{m}\left(\sqrt{\frac{m}{n}}\right)\times\mathbb{S}^{n-m}\left(\sqrt{\frac{n-m}{n}}\right),\qquad 1\leq m\leq \left[\frac{n}{2}\right].$$

Perdomo proved the conjecture when $x$ has antipodal symmetries, and other proofs are known under additional assumptions. The conjecture remains open in general; this paper proves it under the assumption that the first eigenvalue satisfies $\lambda_1<n$.