The joint value-distribution conjecture for phi and sigma fibers

Let $A(n)$ be the number of positive integers $x$ satisfying $\phi(x)=n$, and let $B(n)$ be the number of positive integers $x$ satisfying $\sigma(x)=n$, where $\phi$ is Euler's totient function and $\sigma$ is the sum-of-divisors function. For prescribed integers $k\geqslant 1$ and $l\geqslant 2$, consider simultaneous fibers of these two arithmetic functions.

Joint fiber conjecture. For every $k\geqslant 1$ and $l\geqslant 2$, there are integers $n$ such that

$$A(n)=l\quad\text{and}\quad B(n)=k.$$

This is posed in the paper's section on further problems; the source supplies no resolution, so the problem remains open.