Khabibullin's conjecture for nonnegative logarithmically convex functions

Let $S(t)$ be a nonnegative increasing function on $[0,+\infty)$ that is convex with respect to $\log t$. Let $\rho>1$ and let $n\geq 2$ be an integer. If

$$\int_0^1 S(tx)(1-x^2)^{n-2}\,dx\leq t^{\rho}\qquad (t\geq 0),$$

then

$$\int_0^{+\infty}S(t)\frac{t^{2\rho-1}}{(1+t^{2\rho})^2}\,dt\leq \frac{\pi(n-1)}{2\rho}\prod_{k=1}^{n-1}\left(1+\frac{\rho}{2k}\right).$$

Khabibullin's conjecture. The stated integral implication holds for every such $S$, every $\rho>1$, and every integer $n\geq 2$. This is an extremal integral inequality arising in the Paley problem for plurisubharmonic functions of lower order greater than $1$; the supplied source does not establish whether the conjecture is resolved.