Proposition: Relay attaining the scalar bound

Let $R\ge 1$ be an integer. On the integer line, with $d(i,j)=a|i-j|$, use

Swap the endpoint factors in each block. A factor initially at $0$ is at $tR$ after $t$ ticks, attaining speed $Ra/\tau$. Every finite-horizon relay has a finite-site realization.

Proof. At each tick the blocks partition the line into consecutive intervals of diameter $Ra$. At tick $t$, the factor is at the left endpoint $(t-1)R$ of $B_{t,0}$; the swap sends it to $tR$, the next tick's relay left endpoint.

For horizon $T$, choose a finite interval containing $B_{t,0}$ for every $1\le t\le T$. At each tick retain the partition blocks wholly contained in that interval and replace the clipped boundary portions by singleton blocks. The relay blocks are unchanged. No infinite tensor-product state is needed. The transported factor may be entangled with a reference.