Conjecture on Euler liars and Carmichael or Sophie Germain pseudoprimes

Let $n$ be an odd squarefree composite integer, and let Euler liars in $\mathbb{Z}_n^*$ mean the elements $a$ satisfying the Euler probable-prime congruence $a^{(n-1)/2}\equiv \left(\frac{a}{n}\right)\pmod n$. Euler-liar count conjecture. If $n$ has $\varphi(n)/2$ Euler liars in $\mathbb{Z}_n^*$, then $n$ is a Carmichael number; and if it has $\varphi(n)/4$ Euler liars in $\mathbb{Z}_n^*$, then $n$ is either a Carmichael number or a Sophie Germain pseudoprime. The claim proposes a classification of odd squarefree composites with especially many Euler liars; its resolution is left as future work in the source.