The binary-prefix characterization of as a generalized eigenvalue of LCM and GCD matrices

Let $T=\{1,\dots,n\}$ with $n>3$. Write ${\bf L}_T$ and ${\bf G}_T$ for the LCM and GCD matrices indexed by $T$, respectively, and call $\lambda$ a g-eigenvalue of ${\bf L}_T$ to ${\bf G}_T$ when it is a generalized eigenvalue of the pair. Binary-prefix conjecture. The number $-1$ is a g-eigenvalue of ${\bf L}_T$ to ${\bf G}_T$ if and only if the binary representation of $n$ begins with $10$. This conjecture is motivated by computer experiments for $1\le n\le 1000$ and matches the description of OEIS sequence A004754, after omitting its first term; a proof of the asserted characterization is not supplied here.