Anyone recognize this as a special type of multi-commodity flow problem?
I think you can solve this as a single shortest path problem with respect to arc weights
$$\overline{w}_{ij}=\min\{-w_{ijp}\ :\ p\in\{1,\ldots,m\}\}.$$
For every arc $(i,j)$ fix some $p(i,j)\in\{1,\ldots,m\}$ with $-w_{ijp(i,j)}=\overline{w}_{ij}$. If you have a shortest path $\pi$ with respect to $\overline{w}$, you can put
$$y_{ijp}=\begin{cases} m &amp; \text{if }(i,j)\in\pi\text{ and }p=p(i,j),\\ 0 &amp; \text{otherwise.}\end{cases}$$
For the $l$- and $z$-variables you do the following. 

If $(1,j)\in\pi$ is the arc leaving 1 then
$$z_{1q}=\begin{cases}m-1 &amp; \text{for } q=p(1,j)\\ 0 &amp; \text{for } q\neq p(1,j)\end{cases}$$
$$l_{1q}=\begin{cases}1 &amp; \text{for } q\neq p(1,j)\\ 0 &amp; \text{for } q= p(1,j)\end{cases}$$  

If $(i,n)\in\pi$ is the arc entering $n$ then
$$z_{nq}=\begin{cases}0 &amp; \text{for } q=p(i,n)\\ 1 &amp; \text{for } q\neq p(i,n)\end{cases}$$
$$l_{1q}=\begin{cases}0 &amp; \text{for } q\neq p(i,n)\\ m-1 &amp; \text{for } q= p(i,n)\end{cases}$$  

If $(j,i)$, $(i,j')$ are two consecutive arcs of $\pi$ then you put $l_{ip}=z_{ip}=0$ if $p(j,i)=p(i,j')$, and otherwise
$$z_{iq}=
\begin{cases} 
m &amp; \text{for } q=p(i,j')\\ 
0 &amp; \text{for } q\neq p(i,j')
\end{cases}$$
$$l_{iq}=
\begin{cases}
m &amp; \text{for } q=p(j,i)\\ 
0 &amp; \text{for } q\neq p(j,i)
\end{cases}$$  

And LP duality tells you that this is optimal.