Marques–Trojovský conjecture on the Bonse inequality threshold function

Let $p_n$ denote the $n$-th prime, let $\pi(x)$ be the prime-counting function, and for $x\in\mathbb{R}$ define

$$k(n,x)=n-\pi(n)+\frac{\pi(n)}{\pi(\log n)}-x\,\pi(\pi(n)).$$

For $x\in\mathbb{R}$, let $\Psi(x)$ be the minimum, when it exists, of the integers $m\geq 8$ such that

$$p_1p_2\cdots p_n>p_{n+1}^{k(n,x)}\quad\text{for all }n\geq m.$$

Marques–Trojovský conjecture. The function $\Psi(x)$ satisfies: (i) $\Psi(x)=21$ for all $x\in[0.9,1.3]$; (ii) $\Psi(x)=149$ for all $x\in[0.5,0.8]$; (iii) $\Psi(x)=59{,}875$ for all $x\in[0.3,0.4]$; (iv) $\Psi(0.2)=442{,}414$; and (v) $\Psi(0.1)=24{,}154{,}953$. These values refine the known piecewise-constant behavior of the threshold function and provide explicit numerical information about continuous Bonse-type inequalities; the source does not state whether the conjecture has been resolved.