On a conjecture about an integral criterion for oscillation (Q2716325)

Inspirated by the conjecture of \textit{I. T. Kiguradze} and \textit{T. A. Chanturiya} [Asymptotic properties of solutions of nonautonomous ordinary differential equations. (Russian. English summary) Moskva: Nauka (1990; Zbl 0719.34003), p. 29, problem 1.14], the authors formulate the following, more detailed conjecture: NEWLINENEWLINENEWLINEIf \((-1)^{n-k}p(t)\leq 0\) and NEWLINE\[NEWLINE\int ^{+\infty }t^{n-1}[|p(t)|-\frac {M_{k,n}}{t^n}] dt =+\infty ,NEWLINE\]NEWLINE with \(M_{k,n}=\max \{|\lambda (\lambda -1)\cdots (\lambda -n+1)|:\lambda \in [k-1,k]\}\), then the equation \(y^{(n)}+p(t)y=0\) cannot be \((k, n-k)\)-disfocal on any ray \(]a, +\infty [\). NEWLINENEWLINENEWLINEThis conjecture is verified for the cases \(n=2\), \(n=3\) and \(p(t)>0\), \(n=3\) and \(p(t)<0\), \(n=4\) and \(p(t)<0\).