Continued fraction normality is not preserved along arithmetic progressions

It is well known that if $0.a_1{a}_2{a}_3\dots$ is the base-$b$ expansion of a number normal to base-$b$, then the numbers $0.a_k{a}_{m+k}{a}_{2m+k}\dots$ for $m\ge 2$, $k\ge 1$ are all normal to base-$b$ as well. In contrast, given a continued fraction expansion $\langle a_1,a_2,a_3,\dots\rangle$ that is normal (now with respect to the continued fraction expansion), we show that for any integers $m\ge 2$, $k\ge 1$, the continued fraction $⟨a_k,a_{m+k},a_{2m+k},a_{3m+k},\dots ⟩$ will never be normal.