The infinitude of composite almost-prime numbers

A positive integer $n$ is almost-prime if it is square-free and satisfies $n\mid T_n(x)$ for every integer $x$, where

$$T_n(x)=x^{d_1}+x^{d_2}+\dots+x^{d_k}-kx$$

and $1=d_1<d_2<\dots<d_k=n$ are all divisors of $n$. Infinitude conjecture. There are infinitely many composite almost-prime numbers. Every almost-prime is either prime or a Carmichael number, so this conjecture asks for an infinite special subclass of the composite Carmichael numbers. Its resolution is not specified in the source.