Is it known whether the possible number of prime factors of a Carmichael-number is bounded?

Question

Let $n(C)$ be the number of prime factors of the Carmichael-number $C$.

I Conjecture $\lim sup_{C\rightarrow \infty} n(C)=\infty$

In other words, the sequence $n(C)$, $C$ running over the Carmichael-numbers, is unbounded.

I learnt that Dickson's conjecture implies that this is the case. There are arbitary long strictly increasing vectors $v_1,...,v_n$ ($n\ge 3$) with $\sum_{j=1}^n \frac{1}{v_j}=1$. If $L\ :=\ lcm(v_1,...,v_n)$, then $\prod_{j=1}^n (\frac{L^2}{x_j}\times m+1)$ is a Carmichael-number if $\frac{L^2}{x_j}\times m+1$ is prime for $j=1,...,n$, and Dickson's conjecture implies that such a number $m$ always exists.

I also learnt that it is not known, whether there are infinite many Carmichael numbers with $k$ prime factors for any fixed number $k\ge 3$. But maybe my conjecture can be proven (or disproven).

Answer 1

I don't believe this is currently known. Sequence A006931 references Alford, Grantham, Hayman, & Shallue who, among other things, construct a Carmichael number with 10,333,229,505 prime factors, but don't prove that there are Carmichael numbers with arbitrarily many prime factors.

Answer 2

It turns out that your first question has been answered in the affirmative. It was proved in 1994 that there are Carmichael numbers with arbitrarily many prime factors—see page 708 of Alford, Granville, and Pomerance's paper.