The unique-zero conjecture for the truncated Riemann kernel

Let $n\in\mathbb{N}$ with $n\geqslant 2$. Let $\Phi_n(t)$ be the truncated kernel considered above, let $\omega_n=(1/2)\log n$, and let $\tau_n$ denote its smallest positive zero. Unique-zero conjecture. For each $n\in\mathbb{N}$, $n\geqslant 2$, the function $\Phi_n(t)$ on $[0,\infty)$ has only one zero, $\tau_n$, in the interval

$$(\omega_{n+1},\omega_{n+2})\subset [0,\infty).$$

Moreover,

$$\Phi_n(0\leqslant t<\tau_n)>0,\qquad \Phi_n(\tau_n<t<\infty)<0.$$

This conjecture describes the sign pattern of the truncated kernel and the location of its first positive zero; the paper reports numerical evidence for $n\in[2,12]$, while no proof or resolution is supplied here.