The reduced plane-curve sigma-four conjecture

For a singular plane curve $C\subset\mathbb{P}^2$ of degree $d$ with singular-point multiplicities $m_1,\ldots,m_r$, define

$$\sigma_k(C):=d^2-\sum_{i=1}^r m_i^2+kr.$$

Reduced sigma-four conjecture. For any reduced singular complex curve $C\subset\mathbb{P}^2$, one has

$$\sigma_4(C)>0.$$

The conjecture is equivalent in the paper to a negative answer to the question of whether a reduced singular complex plane curve can have $H$-constant at most $-4$. Reducible curves with $H$-constant close to $-4$ are known, but the source reports no curve with value at most $-4$ and gives no proof of the conjecture.