  Recently, a conjecture on the linear complexity of a new class of generalized
cyclotomic binary sequences of period $p^r$ was proposed by Z. Xiao et al.
(Des. Codes Cryptogr., DOI 10.1007/s10623-017-0408-7). Later, for the case $f$
being the form $2^r$ with $r\ge 1$, Vladimir Edemskiy proved the conjecture
(arXiv:1712.03947). In this paper, under the assumption of $2^{p-1} \not\equiv
1 \bmod p^2$ and $\gcd(\frac{p-1}{{\rm {ord}}_{p}(2)},f)=1$, the conjecture
proposed by Z. Xiao et al. is proved for a general $f$ by using the Euler
quotient. Actually, a generic construction of $p^r$-periodic binary sequence
based on the generalized cyclotomy is introduced in this paper, which admits a
flexible support set and includes Xiao's construction as a special case, and
then an efficient method to compute the linear complexity of the sequence by
the generic construction is presented, based on which the conjecture proposed
by Z. Xiao et al. could be easily proved under the aforementioned assumption.
