Generalized Carmichael counting-function symmetry conjecture

For an integer $k$, let $C_k$ be the set of positive integers $n$ such that $n > \max(k,0)$ and $a^{n-k+1} \equiv a \pmod{n}$ for all integers $a$. For squarefree $k>0$, define the generalized Carmichael numbers by

$$N_k=C_k\setminus\left\{kp:p\equiv 1\pmod{\frac{\lambda(k)}{\gcd(\lambda(k),k)}}\right\},$$

and set $N_k=C_k$ for the other values of $k$. Let

$$N_k(X)=\lvert N_k\cap(0,X]\rvert.$$

Generalized Carmichael symmetry conjecture. For all integers $k$,

$$\lim_{X\to\infty}\frac{N_{-k}(X)}{N_k(X)}=1.$$

The conjecture is based on numerical comparisons of the counting functions and concerns the asymptotic relationship between generalized Carmichael numbers for opposite parameters. The source gives no resolution, so it remains open.