Teissier's conjecture on Arnold exponents and polar invariants

Let $X$ be a smooth complex $n$-dimensional algebraic variety, let $f\in\operatorname{\mathcal O}_X(X)$ be nonzero, and let $P\in X$ be a point in the zero-locus of $f$ such that $f$ has an isolated singularity at $P$. If $H_1,\ldots,H_{n-1}$ are hypersurfaces in $X$ passing through $P$, such that each $\Lambda_i:=H_1\cap\ldots\cap H_i$ is smooth at $P$ of dimension $n-i$, and such that $f_i:=f\vert_{\Lambda_i}$ has an isolated singularity at $P$, then Teissier's conjecture.

$$\sigma_P(f)\geq\frac{1}{1+\theta_P(f)}+\frac{1}{1+\theta_P(f_1)}+\ldots+\frac{1}{1+\theta_P(f_{n-1})}.$$

Here $\theta_P(f)$ is Teissier's invariant and $\sigma_P(f)$ is the Arnold exponent of $f$ at $P$. The paper proves this conjecture in the case of log canonical thresholds, providing the stated lower bound through successive smooth hyperplane sections.