Conjecture on the Kertész-line asymptotic for the two-dimensional Ising model

Consider the two-dimensional Ising model and let $h_c(p)$ denote the Kertész line as a function of the parameter $p$, with $p_c=p_c(2,0)$ the zero-field critical point. Kertész-line asymptotic conjecture. In the limit $p \to p_c$, there is a constant $c>0$ such that

$$h_c(p) \sim c (p-p_c)^{\frac{15}{8}}.$$

This conjecture proposes the critical asymptotic behaviour of the Kertész line near the zero-field critical point. The supplied text presents the determination of the asymptote as an open problem and gives no resolution of this conjecture.