The limiting-distribution conjecture for totient multiplicities

Let $\phi$ be Euler's totient function. For $x\geqslant 1$, let $V(x)$ be the number of totients at most $x$, and for an integer $k\geqslant 1$, let $V_k(x)$ be the number of totients at most $x$ having exactly $k$ preimages under $\phi$. Limiting-distribution conjecture. For every integer $k\geqslant 2$, there exists a constant $C_k$ such that

$$\lim_{x\to\infty}\frac{V_k(x)}{V(x)}=C_k.$$

The preceding theorem proves only that whenever multiplicity $k$ is possible, $V_k(x)$ is a positive proportion of $V(x)$ up to a factor depending on a witnessing totient. The existence of the individual limiting proportions $C_k$ remains conjectural; numerical data in the source suggests a possible order of magnitude $C_k\asymp 1/k^2$.