Local intersection multiplicity conjecture for cuspidal curves and their 2-Hessians

Let $C$ be a cuspidal plane algebraic curve and let $H_2$ be its 2-Hessian curve. At a point $p\in C$, write $m$ for the multiplicity of $C$ at $p$, $\delta$ for its delta invariant, $l$ for the intersection multiplicity of $C$ with its tangent at $p$, and $c$ for its intersection multiplicity with an osculating conic at $p$. Local intersection multiplicity conjecture. The intersection multiplicity $(H_2\cdot C)_p$ is determined as follows. If $l\neq 2m$, then

$$(H_2\cdot C)_p=24\delta+3m+3l-12.$$

If $l=2m$, then

$$(H_2\cdot C)_p=24\delta+7m+c-12.$$

The conjecture gives a local geometric interpretation of the correction terms in the global Bézout formula for the intersection of a cuspidal curve with its 2-Hessian. It was verified for all rational cuspidal curves of degrees $4$ and $5$; the corresponding analogous result for Hessian curves is known, while the general case remains open.