Khabibullin's integral inequality conjecture

Let $\lambda$ be a positive real number and let $n$ be an integer with $n\geqslant 2$. Let $S(x)$ be a nonnegative nondecreasing function on $[0,+\infty)$. If

$$\int\limits^{\,1}_0\!S(tx)\,(1-x^2)^{n-2} \,x\,dx\leqslant t^{\lambda}$$

for every $0\leqslant t<+\infty$, then Khabibullin's conjecture.

$$\int\limits^{+\infty}_0\!\!S(t)\,\frac{t^{2\lambda-1}}{(1+t^{2\lambda})^2} \,dt\leqslant\frac{\pi\,(n-1)}{2\lambda}\,\prod^{n-1}_{k=1}\Bigl(1 +\frac{\lambda}{2k}\Bigr).$$

This conjecture is the subject of the paper's analysis of Sharipov's purported counterexample; the supplied status evidence marks the conjecture as disproved.