Does the conjugate symmetry property of the Fourier series coefficients only hold for purely real input signals?
Yes, it only holds for real-valued signals. If you consider the complex Fourier coefficients of the \$T\$-periodic function \$f(t)\$

$$c_n=\frac{1}{T}\int_0^Tf(t)e^{-j2\pi nt/T}dt\tag{1}$$

then the complex conjugated coefficients are given by

$$c_n^*=\frac{1}{T}\int_0^Tf^*(t)e^{j2\pi nt/T}dt\tag{2}$$

and, consequently,

$$c_{-n}^*=\frac{1}{T}\int_0^Tf^*(t)e^{-j2\pi nt/T}dt\tag{3}$$

which is only equivalent to (1) if \$f(t)=f^*(t)\$, i.e. if \$f(t)\$ is real-valued.