The existence of composite weakly almost-prime and almost-prime numbers

For each positive integer $n$, let

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

where $1=d_1<d_2<\dots<d_k=n$ are all divisors of $n$. A positive integer $n$ is weakly almost-prime if $n\mid T_n(x)$ for every integer $x$; if it is also square-free, it is almost-prime. Existence conjecture. There are no composite weakly almost-prime numbers greater than four and no composite almost-prime numbers. The source phrases this as a question, so its resolution is not specified here.