Extension conjecture for Nim sequences under the recurrence $b_i=(m+1)b_{i-1}-(m-1)b_{i-2}$

Let $(a_i)_{i=0}^n$ with $n\geq2$ be a finite representing sequence with representation word $w$, and let $m$ be an integer with $n\geq2$. Define $(b_i)$ by

$$b_i=\begin{cases} a_i & i\leq n\\ (m+1)b_{i-1}-(m-1)b_{i-2} & i>n,\end{cases}$$

and let $v$ be the representation word of $(b_i)$. Extension conjecture. If $w$ is the Nim sequence of $T((a_i))$ and

$$a_n=(m+1)a_{n-1}-(m-1)a_{n-2},$$

then $v$ is the Nim sequence of $T((b_i))$. The conjecture concerns extending a Nim-sequence-producing representing sequence by an eventual recurrence with parameter $m$; its resolution is not given in the supplied text.